Trigonometric Systems [version 0.50]

Trigonometric Systems

(technical report)

Daniel J. Greenhoe

title: Trigonometric Systems document type: technical report author: Daniel J. Greenhoe url: https://www.researchgate.net/profile/Daniel_Greenhoe/publications

Copyright © 2017 Daniel J. Greenhoe All rights reserved

This text was typeset using XƎLATEX, which is part of the TEXfamily of typesetting engines, which is arguably the greatest development since the Gutenberg Press. Graphics were rendered using the pstricks and related packages, and LATEX graphics support. The main roman, italic, and bold font typefaces used are all from the Heuristica family of typefaces (based on the Utopia typeface, released by Adobe Systems Incorporated). The math font is XITS from the XITS font project. The font used in quotation boxes is adapted from Zapf Chancery Medium Italic, originally from URW++ Design and Development Incorporated. The font used for the text in the title is Adventor (similar to Avant-Garde) from the TEX-Gyre Project. The font used for the ISBN in the footer of individual pages is Liquid Crystal (Liquid Crystal) from FontLab Studio. The Latin handwriting font is Lavi from the Free Software Foundation. The traditional Chinese font used for the author's Chinese name is 王漢宗中仿宋繁.1 Chinese glyphs appearing elsewhere in the text are from the font 王漢宗中明體繁.2 The ship appearing throughout this text is loosely based on the Golden Hind, a sixteenth century English galleon famous for circumnavigating the globe.3

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pinyin: Wáng Hàn Zōng Zhōng Fǎng Sòng Fán; translation: Hàn Zōng Wáng's Medium-weight Sòng-style Traditional Characters; literal: 王漢宗∼font designer's name; 中∼medium; 仿∼to imitate; 宋∼Sòng (a dynasty); 繁∼traditional 2 pinyin: Wáng Hàn Zōng Zhōng Mińg Tǐ Fán; translation: Hàn Zōng Wáng's Medium-weight Mińg-style Traditional Characters; literal: 王漢宗∼font designer's name; 中∼medium; 明∼Mińg (a dynasty); 體∼style; 繁∼traditional 3 📘 Paine (2000) page 63 ⟨Golden Hind⟩

❝Here, on the level sand,

Between the sea and land, What shall I build or write Against the fall of night? ❞

❝Tell me of runes to grave

That hold the bursting wave, Or bastions to design For longer date than mine. ❞

Alfred Edward Housman, English poet (1859–1936)

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❝The uninitiated imagine that one must await inspiration in order to

create. That is a mistake. I am far from saying that there is no such thing as inspiration; quite the opposite. It is found as a driving force in every kind of human activity, and is in no wise peculiar to artists. But that force is brought into action by an effort, and that effort is work. Just as appetite comes by eating so work brings inspiration, if inspiration is not discernible at the beginning. ❞ Igor Fyodorovich Stravinsky (1882–1971), Russian-born composer

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❝As

I think about acts of integrity and grace, I realise that there is nothing in my knowledge to compare with Frege's dedication to truth. His entire life's work was on the verge of completion, much of his work had been ignored to the benefit of men infinitely less capable, his second volume was about to be published, and upon finding that his fundamental assumption was in error, he responded with intellectual pleasure clearly submerging any feelings of personal disappointment. It was almost superhuman and a telling indication of that of which men are capable if their dedication is to creative work and knowledge instead of cruder efforts to dominate and be known.❞ Bertrand Russell (1872–1970), British mathematician, in a 1962 November 23 letter to Dr. van Heijenoort.

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📘 Housman (1936), page 64 ⟨“Smooth Between Sea and Land”⟩, 📘 Hardy (1940) ⟨section 7⟩ http://en.wikipedia.org/wiki/Image:Housman.jpg 📘 Ewen (1961), page 408, 📘 Ewen (1950) http://en.wikipedia.org/wiki/Image:Igor_Stravinsky.jpg 📘 Heijenoort (1967), page 127 http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Russell.html Trigonometric Systems

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SYMBOLS

❝rugula

XVI. Quae vero praesentem mentis attentionem non requirunt, etiamsi ad conclusionem necessaria sint, illa melius est per brevissimas notas designare quam per integras figuras: ita enim memoria non poterit falli, nec tamen interim cogitatio distrahetur ad haec retinenda, dum aliis deducendis incumbit.❞

❝Rule XVI. As for things which do not

require the immediate attention of the mind, however necessary they may be for the conclusion, it is better to represent them by very concise symbols rather than by complete figures. It will thus be impossible for our memory to go wrong, and our mind will not be distracted by having to retain these while it is taken up with deducing other matters.❞

René Descartes (1596–1650), French philosopher and mathematician

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❝In

signs one observes an advantage in discovery which is greatest when they express the exact nature of a thing briefly and, as it were, picture it; then indeed the labor of thought is wonderfully diminished.❞ Gottfried Leibniz (1646–1716), German mathematician,

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Symbol list symbol numbers: ℤ 𝕎

description integers whole numbers

… , −3, −2, −1, 0, 1, 2, 3, … 0, 1, 2, 3, …

…continued on next page… 7

quote: 📘 Descartes (1684a) ⟨rugula XVI⟩, translation: 📘 Descartes (1684b) ⟨rule XVI⟩, image: Frans Hals (circa 1650), http://en.wikipedia.org/wiki/Descartes, public domain 8 quote: 📘 Cajori (1993) ⟨paragraph 540⟩, image: http://en.wikipedia.org/wiki/File:Gottfried_Wilhelm_von_Leibniz.jpg, public domain

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Symbol list

symbol description ℕ natural numbers 1, 2, 3, … ⊣ ℤ non-positive integers … , −3, −2, −1, 0 − ℤ negative integers … , −3, −2, −1 ℤo odd integers … , −3, −1, 1, 3, … ℤe even integers … , −4, −2, 0, 2, 4, … 𝑚 with 𝑚 ∈ ℤ and 𝑛 ∈ ℤ⧵0 ℚ rational numbers 𝑛 ℝ real numbers completion of ℚ ⊢ ℝ non-negative real numbers [0, ∞) ⊣ ℝ non-positive real numbers (−∞, 0] + ℝ positive real numbers (0, ∞) ℝ− negative real numbers (−∞, 0) ∗ ℝ extended real numbers ℝ∗ ≜ ℝ ∪ {−∞, ∞} ℂ complex numbers 𝔽 arbitrary field (often either ℝ or ℂ) ∞ positive infinity −∞ negative infinity 𝜋 pi 3.14159265 … relations: Ⓡ relation ? relational and 𝑋×𝑌 Cartesian product of 𝑋 and 𝑌 (△, ▽) ordered pair absolute value of a complex number 𝑧 |𝑧| = equality relation ≜ equality by definition → maps to ∈ is an element of ∉ is not an element of 𝓓(Ⓡ) domain of a relation Ⓡ 𝓘(Ⓡ) image of a relation Ⓡ 𝓡(Ⓡ) range of a relation Ⓡ 𝓝 (Ⓡ) null space of a relation Ⓡ set relations: ⊆ subset ⊊ proper subset ⊇ super set ⊋ proper superset ⊈ is not a subset of ⊄ is not a proper subset of operations on sets: 𝐴∪𝐵 set union 𝐴∩𝐵 set intersection 𝐴 △𝐵 set symmetric difference 𝐴⧵𝐵 set difference 𝖼 𝐴 set complement set order |⋅| 𝟙𝐴 (𝑥) set indicator function or characteristic function logic: 1 “true” condition …continued on next page… CC

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Daniel J. Greenhoe

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symbol description 0 “false” condition ¬ logical NOT operation ∧ logical AND operation ∨ logical inclusive OR operation ⊕ logical exclusive OR operation ⟹ “implies”; “only if” ⟸ “implied by”; “if” ⟺ “if and only if”; “implies and is implied by” ∀ universal quantifier: “for each” ∃ existential quantifier: “there exists” order on sets: ∨ join or least upper bound ∧ meet or greatest lower bound ≤ reflexive ordering relation “less than or equal to” ≥ reflexive ordering relation “greater than or equal to” < irreflexive ordering relation “less than” > irreflexive ordering relation “greater than” measures on sets: order or counting measure of a set 𝑋 |𝑋 | distance spaces: 𝖽 metric or distance function linear spaces: vector norm ‖⋅‖ operator norm ⦀⋅⦀ ⟨△ | ▽⟩ inner-product 𝗌𝗉𝖺𝗇(𝙑 ) span of a linear space 𝙑 algebras: ℜ real part of an element in a ∗-algebra ℑ imaginary part of an element in a ∗-algebra set structures: 𝑻 a topology of sets 𝑹 a ring of sets 𝑨 an algebra of sets ∅ empty set 𝑋 𝟚 power set on a set 𝑋 sets of set structures: 𝒯 (𝑋) set of topologies on a set 𝑋 ℛ(𝑋) set of rings of sets on a set 𝑋 𝒜(𝑋) set of algebras of sets on a set 𝑋 classes of relations/functions/operators: 𝟚𝑋𝑌 set of relations from 𝑋 to 𝑌 𝑋 𝑌 set of functions from 𝑋 to 𝑌 𝒮j (𝑋, 𝑌 ) set of surjective functions from 𝑋 to 𝑌 ℐj (𝑋, 𝑌 ) set of injective functions from 𝑋 to 𝑌 ℬj (𝑋, 𝑌 ) set of bijective functions from 𝑋 to 𝑌 ℬ(𝙓 , 𝙔 ) set of bounded functions/operators from 𝙓 to 𝙔 ℒ(𝙓 , 𝙔 ) set of linear bounded functions/operators from 𝙓 to 𝙔 𝒞(𝙓 , 𝙔 ) set of continuous functions/operators from 𝙓 to 𝙔 specific transforms/operators: …continued on next page… rd

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symbol 𝐅̃ 𝐅̂ 𝐅̆ 𝐙 ̃ 𝖿(𝜔) ̆ 𝗑(𝜔) ̂ 𝗑(𝑧)

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description Fourier Transform operator (Definition 3.2 page 41) Fourier Series operator (Definition 5.1 page 65) Discrete Time Fourier Series operator (Definition 6.1 page 69) Z-Transform operator (Definition C.4 page 91) Fourier Transform of a function 𝖿(𝑥) ∈ 𝙇𝟤ℝ Discrete Time Fourier Transform of a sequence ⦅𝑥𝑛 ∈ ℂ⦆𝑛∈ℤ Z-Transform of a sequence ⦅𝑥𝑛 ∈ ℂ⦆𝑛∈ℤ


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SYMBOL INDEX

⟨△ | ▽⟩, 112 ‖⋅‖, 106 𝐶 ∗ , 85 𝑃 , 39 𝑇𝑛 (𝑥), 24 𝑇𝑛 , 24 ℂ, 49 𝖣𝑛 , 30 ℑ, 83 𝖩𝑛 , 39 𝖪𝑛 , 37 ℚ, 55 ℝ, 49 ℜ, 83

𝖵𝑛 , 39 ℬ(𝙓 , 𝙔 ), 109 𝑌 𝑋 , 49 ℒ(ℂ, ℂ), 60 𝙔 𝙓 , 103 ⋆, 44 ✶, 91 𝖼𝗈𝗌(𝑥), 3 𝖼𝗈𝗌, 13 exp(𝑖𝑥), 8 ≥, 127 ∗, 82 𝛿𝑛̄ , 102 ⦀⋅⦀, 106

𝐃𝛼 , 50 𝐃, 50 𝐃∗ , 53 𝐅,̂ 65 𝐅̂ ∗ , 67 𝐅̂ −1 , 66 𝐅,̃ 42 𝐈, 103 𝐓𝜏 , 50 𝐓, 50 𝐓 ∗ , 53 𝐙, 91 𝑟, 81 𝜌, 81

𝜎, 81 ⦅𝜅𝑛 ⦆𝑛∈ℤ , 37 𝑋, 49 𝑌 , 49 𝟙, 46 𝗌𝗂𝗇(𝑥), 3 𝗌𝗂𝗇, 13 ℂℂ , 49 𝙇𝟤ℝ , 87 𝙇𝟤(ℝ,ℬ,𝜇) , 87 𝙋𝙒𝜎2 , 61 ℝℝ , 49 tan, 13 𝑤𝘕 , 31

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CONTENTS

Front matter Front cover . . . . . . . . Title page . . . . . . . . . Copyright and typesetting Quotes . . . . . . . . . . Symbol list . . . . . . . . Symbol index . . . . . . . Contents . . . . . . . . .

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i . i . v . vi . vii . ix . xiii . xv

1 Trigonometric Functions 1.1 Definition Candidates . . . . 1.2 Definitions . . . . . . . . . . . 1.3 Basic properties . . . . . . . 1.4 The complex exponential . . 1.5 Trigonometric Identities . . . 1.6 Planar Geometry . . . . . . . 1.7 The power of the exponential

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1 1 3 3 8 10 16 17

2 Trigonometric Polynomials 2.1 Trigonometric expansion . 2.2 Trigonometric reduction . 2.3 Spectral Factorization . . 2.4 Dirichlet Kernel . . . . . . 2.5 Trigonometric summations 2.6 Summability Kernels . . .

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3 Fourier Transform 3.1 Definitions . . . . . . . 3.2 Operator properties . 3.3 Convolution . . . . . . 3.4 Real valued functions 3.5 Examples . . . . . . .

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41 41 42 44 45 46

4 Transversal Operators 4.1 Families of Functions . . . . . . . . . 4.2 Definitions and algebraic properties 4.3 Linear space properties . . . . . . . 4.4 Inner-product space properties . . . 4.5 Normed linear space properties . . . 4.6 Fourier transform properties . . . . . 4.7 Examples . . . . . . . . . . . . . . . 4.8 Cardinal Series and Sampling . . . . 4.8.1 Cardinal series basis . . . . . 4.8.2 Sampling . . . . . . . . . . .

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CONTENTS

5 Fourier Series 5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Inverse Fourier Series operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Fourier series for compactly supported functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 66 68

6 Discrete Time Fourier Transform 6.1 Definition . . . . . . . . . . . 6.2 Properties . . . . . . . . . . . 6.3 Derivatives . . . . . . . . . . 6.4 Frequency Response . . . .

69 69 69 75 77

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Appendices

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A Normed Algebras A.1 Algebras . . . . . A.2 Star-Algebras . . A.3 Normed Algebras A.4 C* Algebras . . .

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B Calculus

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C Operations on Sequences C.1 Z-transform . . . . . . . . . . C.2 Zero Locations . . . . . . . . C.3 Rational polynomial operators C.4 Sample rate conversion . . . C.5 Filter Banks . . . . . . . . . .

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91 91 92 93 95 96

D Operators on Linear Spaces D.1 Operators on linear spaces . . . . . . . . D.1.1 Operator Algebra . . . . . . . . . . D.1.2 Linear operators . . . . . . . . . . D.2 Operators on Normed linear spaces . . . D.2.1 Operator norm . . . . . . . . . . . D.2.2 Bounded linear operators . . . . . D.2.3 Adjoints on normed linear spaces . D.3 Operators on Inner-product spaces . . . . D.3.1 General Results . . . . . . . . . . D.3.2 Operator adjoint . . . . . . . . . . D.4 Special Classes of Operators . . . . . . . D.4.1 Projection operators . . . . . . . . D.4.2 Self Adjoint Operators . . . . . . . D.4.3 Normal Operators . . . . . . . . . D.4.4 Isometric operators . . . . . . . . . D.4.5 Unitary operators . . . . . . . . . . D.5 Operator order . . . . . . . . . . . . . . .

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version 0.50


CHAPTER 1 TRIGONOMETRIC FUNCTIONS

1.1 Definition Candidates There are several ways of defining the sine and cosine functions, including the following:1

1. Planar geometry: Trigonometric functions are traditionally introduced as they have come to us historically—that is, as related to the parameters of triangles.2 𝑦

𝑟 𝑥 𝑥

𝖼𝗈𝗌𝑥 ≜ 𝗌𝗂𝗇𝑥 ≜

𝑥 𝑦𝑟 𝑟

2. Complex exponential: The cosine and sine functions are the real and imaginary parts of the complex exponential such that3 𝖼𝗈𝗌𝑥 ≜ ℜ𝑒𝑖𝑥 𝗌𝗂𝗇𝑥 ≜ ℑ𝑒𝑖𝑥 ∞

3. Polynomial: Let

𝘕

∑ 𝑛=0

𝑥𝑛 ≜ lim

𝘕 →∞ ∑ 𝑛=0

𝑥𝑛 in some topological space. The sine and cosine functions

can be defined in terms of Taylor expansions such that4 ∞

(−1)𝑛 2𝑛 𝖼𝗈𝗌(𝑥) ≜ 𝑥 ∑ (2𝑛)! 𝑛=0

= 1−

𝑥2 𝑥4 𝑥6 + − +⋯ 2 4! 6!

∞

𝗌𝗂𝗇(𝑥) ≜ 1

(−1)𝑛 𝑥3 𝑥5 𝑥7 𝑥2𝑛+1 = 𝑥 − + − +⋯ ∑ (2𝑛 + 1)! 3! 5! 7! 𝑛=0

The term sine originally came from the Hindu word jiva and later adapted to the Arabic word jiba. Abrabic-Latin translator Robert of Chester apparently confused this word with the Arabic word jaib, which means “bay” or “inlet”— thus resulting in the Latin translation sinus, which also means “bay” or “inlet”. Reference: 📘 Boyer and Merzbach (1991) page 252 2 📘 Abramowitz and Stegun (1972), page 78 3 📘 Euler (1748) 4 📘 Rosenlicht (1968), page 157, 📘 Abramowitz and Stegun (1972), page 74

page 2

Daniel J. Greenhoe

CHAPTER 1. TRIGONOMETRIC FUNCTIONS

∞

4. Product of factors: Let

∏

𝘕

𝑥𝑛 ≜ lim

𝘕 →∞ ∏ 𝑛=0

𝑛=0

𝑥𝑛 in some topological space. The sine and cosine

functions can be defined in terms of a product of factors such that5 ∞

⎡ 𝑥 ⎢1 − 𝖼𝗈𝗌(𝑥) ≜ ∏⎢ (2𝑛 − 1) 𝜋2 ) ( 𝑛=1 ⎣

2

∞

⎤ ⎥ ⎥ ⎦

𝗌𝗂𝗇(𝑥) ≜ 𝑥

𝑥 2 1−( ) ∏[ 𝑛𝜋 ] 𝑛=1

5. Partial fraction expansion: The sine function can be defined in terms of a partial fraction expansion such that6 𝗌𝗂𝗇(𝑥) ≜

∞

1 1 𝖼𝗈𝗌(𝑥) ≜ 𝗌𝗂𝗇(𝑥) + 2𝑥 2 2) ∑ 𝑥 𝑥 − (𝑛𝜋) ( 𝑛=1 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1 ∞

(−1)𝑛 1 + 2𝑥 ∑ 𝑥2 − (𝑛𝜋)2 𝑥 𝑛=1

cot(𝑥)

6. Differential operator: The sine and cosine functions can be defined as solutions to differen𝗱 tial equations expressed in terms of the differential operator 𝗱𝘅 such that 𝖼𝗈𝗌(𝑥) ≜ 𝖿 (𝑥)

𝗌𝗂𝗇(𝑥) ≜ 𝗀(𝑥)

𝟮

where

𝗱 𝖿 +𝖿 =0 𝗱𝘅𝟮 ⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝖿(0) =1 ⏟⏟⏟

differential equation

1st initial condition

𝗱𝟮

where

𝗀+𝗀=0 𝗱𝘅𝟮 ⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝗀(0) = 0 ⏟⏟⏟

differential equation

1st initial condition

𝗱

𝖿 (0) = 0 [⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝗱𝘅 ] 2nd initial condition

𝗱

𝗀 (0) = 1 [⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝗱𝘅 ] 2nd initial condition

7. Integral operator: The sine and cosine functions can be defined as inverses of integrals of square roots of rational functions such that7 𝖼𝗈𝗌(𝑥) ≜ 𝖿

−1

1

(𝑥) where 𝖿(𝑥) ≜

1 𝖽𝑦 √ ∫ 1 − 𝑦2 𝑥 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ arccos(𝑥)

𝗌𝗂𝗇(𝑥) ≜ 𝗀−1 (𝑥) where 𝗀(𝑥) ≜

𝑥

1 𝖽𝑦 √ ∫ 1 − 𝑦2 0 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ arcsin(𝑥)

For purposes of analysis, it can be argued that the more natural approach for defining harmonic 𝗱 (Definition 1.1 page 3). Support for such an apfunctions is in terms of the differentiation operator 𝗱𝘅 proach includes the following: Both sine and cosine are very easily represented analytically as polynomials with coeffi𝗱 (Theorem 1.1 page 4). cients involving the operator 𝗱𝘅 All solutions of homogeneous second order differential equations are linear combinations of sine and cosine (Theorem 1.3 page 6). Sine and cosine themselves are related to each other in terms of the differentiation operator (Theorem 1.4 page 7). 5

📘 Abramowitz and Stegun (1972), page 75 📘 Abramowitz and Stegun (1972), page 75 7 📘 Abramowitz and Stegun (1972), page 79 6

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1.2. DEFINITIONS

page 3

Daniel J. Greenhoe

The complex exponential function is a solution of a second order homogeneous differential equation (Definition 1.4 page 8). Sine and cosine are orthogonal with respect to an innerproduct generated by an integral operator—which is a kind of inverse differential operator (Section 1.6 page 16).

1.2 Definitions Definition 1.1. 8 Let 𝘾 be the SPACE OF ALL CONTINUOUSLY DIFFERENTIABLE REAL FUNCTIONS and 𝗱 ∈ 𝘾 𝘾 the differentiation operator. 𝗱𝘅 The function 𝖿 ∈ ℂℂ is the cosine function 𝖼𝗈𝗌(𝑥) ≜ 𝖿(𝑥) if

d e f

1.

𝗱𝟮 𝖿 𝗱𝘅𝟮

2. 3.

+𝖿 = 0 𝖿(0) = 1

𝗱 [ 𝗱𝘅 𝖿 ](0)

Definition 1.2.

9

= 0

Let 𝘾 and

(second order homogeneous differential equation)

and

(first initial condition)

and

(second initial condition).

𝗱 𝗱𝘅

∈ 𝘾 𝘾 be defined as in definition of 𝖼𝗈𝗌(𝑥) (Definition 1.1 page 3).

The function 𝖿 ∈ ℂℂ is the sine function 𝗌𝗂𝗇(𝑥) ≜ 𝖿(𝑥) if d e f

1. 2. 3.

𝗱𝟮 𝖿 𝗱𝘅𝟮

+𝖿 = 0 𝖿(0) = 0

𝗱 [ 𝗱𝘅 𝖿 ](0)

= 1

(second order homogeneous differential equation)

and

(first initial condition)

and

(second initial condition).

Definition 1.3. 10 Let 𝜋 (“pi”) be defined as the element in ℝ such that 𝜋 d (1). 𝖼𝗈𝗌 ( 2 ) = 0 and e f (2). 𝜋 > 0 and (3). 𝜋 is the smallest of all elements in ℝ that satisfies (1) and (2).

1.3 Basic properties Lemma 1.1. 11 Let 𝘾 be the SPACE OF ALL CONTINUOUSLY DIFFERENTIABLE REAL FUNCTIONS and 𝗱 ∈ 𝘾 𝘾 the differentiation operator. 𝗱𝘅 𝟮

𝗱 ⟺ { 𝗱𝘅𝟮 𝖿 + 𝖿 = 0} ∞ ∞ 2𝑛+1 2𝑛 ⎧ 𝗱 𝑛 𝑥 𝑛 𝑥 (−1) 𝖿 (0) + (−1) 𝖿(𝑥) = [𝖿](0) ⎪ ∑ ∑ (2𝑛)! [ 𝗱𝘅 ] (2𝑛 + 1)! 𝑛=0 𝑛=0 ⎪ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⎪ even terms odd terms ⎨ 𝗱 𝗱 ⎪ ⎛ 𝖿(0) [ 𝗱𝘅 𝖿 ](0) 3 ⎞ ⎛ 𝖿(0) 4 [ 𝗱𝘅 𝖿 ](0) 5 ⎞ 𝗱 ⎪ 2 𝑥 + 𝑥 ⎟+⎜ 𝑥 + 𝑥 ⎟⋯ = 𝖿(0) + [ 𝗱𝘅 𝖿 ](0)𝑥 − ⎜ ⎪ ( ) ⎜ 2! 3! 5! ⎟ ⎜ 4! ⎟ ⎩ ⎠ ⎝ ⎠ ⎝

l e m

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

8

📘 Rosenlicht (1968) page 157, 📘 Flanigan (1983) pages 228–229 📘 Rosenlicht (1968) page 157, 📘 Flanigan (1983) pages 228–229 10 📘 Rosenlicht (1968) page 158 11 📘 Rosenlicht (1968), page 156, 📘 Liouville (1839) 9



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page 4

Daniel J. Greenhoe 𝗱 𝖿(𝑥). 𝗱𝘅

Let 𝖿 ′ (𝑥) ≜

✎PROOF:


𝗱 𝖿 ′′′ (𝑥) = −[ 𝗱𝘅 𝖿 ](𝑥) 𝟮

𝗱 𝖿 (4) (𝑥) = −[ 𝗱𝘅 𝖿 ](𝑥)

1. Proof that

𝗱𝟮 [ 𝗱𝘅𝟮 𝖿 ](𝑥)

+ 𝖿 (𝑥) = 0 ⟹ 𝖿 (𝑥) =

𝗱 𝖿 (𝑥) = 𝖿 (𝑥) = −[ 𝗱𝘅 𝟮 ]

⎡

𝑛 ⎢ 𝖿(0) 2𝑛 ∑∞ 𝑛=0 (−1) ⎢ (2𝑛)! 𝑥

+

⎣

𝗱 𝖿 (0) ⎤ [ 𝗱𝘅 ] 2𝑛+1 ⎥ 𝑥 : (2𝑛+1)! ⎥

⎦

∞

𝖿 (𝑥) =

𝖿 (𝑛) (0) 𝑛 𝑥 ∑ 𝑛! 𝑛=0

by Taylor expansion 𝗱𝟮

[ 𝗱𝘅𝟮 𝖿 ](0) 2 𝖿 3 (0) 3 𝖿 4 (0) 4 𝖿 5 (0) 5 𝗱 = 𝖿 (0) + [ 𝗱𝘅 𝖿 ](0)𝑥 − 𝑥 − 𝑥 + 𝑥 + 𝑥 −⋯ 2! 3! 4! 5! 𝗱 [ 𝗱𝘅 𝖿 ](0)𝑥

= 𝖿 (0) +

𝗱 𝗱 𝖿(0) 2 [ 𝗱𝘅 𝖿 ](0) 3 𝖿(0) 4 [ 𝗱𝘅 𝖿 ](0) 5 − 𝑥 − 𝑥 + 𝑥 + 𝑥 −⋯ 2! 3! 4! 5!

𝗱 𝖿 ](0) ⎤ ⎡ 𝖿(0) [ 𝗱𝘅 𝑥2𝑛 + 𝑥2𝑛+1 ⎥ = 𝖿 (𝑥) = (−1)𝑛 ⎢ ∑ (2𝑛 + 1)! ⎥ ⎢ (2𝑛)! 𝑛=0 ⎦ ⎣ ∞

2. Proof that

𝗱𝟮 [ 𝗱𝘅𝟮 𝖿 ](𝑥)

+ 𝖿 (𝑥) = 0 ⟸ 𝖿 (𝑥) =

⎡

𝑛 ⎢ 𝖿(0) 2𝑛 ∑∞ 𝑛=0 (−1) ⎢ (2𝑛)! 𝑥

⎣

𝗱𝟮

[ 𝗱𝘅𝟮 𝖿 ](𝑥) =

+

𝗱 𝖿 (0) ⎤ [ 𝗱𝘅 ] 2𝑛+1 ⎥ 𝑥 : (2𝑛+1)! ⎥

⎦

𝗱 𝗱 [𝖿 (𝑥)] 𝗱𝘅 𝗱𝘅

𝗱 𝖿 ](0) ⎡ 𝖿(0) ⎤ [ 𝗱𝘅 𝗱 𝗱 𝑛⎢ 2𝑛 2𝑛+1 ⎥ (−1) 𝑥 + 𝑥 = 𝗱𝘅 𝗱𝘅 ∑ (2𝑛 + 1)! ⎢ (2𝑛)! ⎥ 𝑛=0 ⎣ ⎦ ∞

by right hypothesis

𝗱 𝖿 ](0) (2𝑛 + 1)(2𝑛)[ 𝗱𝘅 ⎡ (2𝑛)(2𝑛 − 1)𝖿 (0) ⎤ = (−1)𝑛 ⎢ 𝑥2𝑛−2 + 𝑥2𝑛−1 ⎥ ∑ (2𝑛)! (2𝑛 + 1)! ⎢ ⎥ 𝑛=1 ⎣ ⎦ ∞

𝗱 [ 𝗱𝘅 𝖿 ](0) 2𝑛−1 ⎤ 𝖿(0) 2𝑛−2 ⎥ 𝑥 + 𝑥 = (−1) ∑ (2𝑛 − 1)! ⎥ ⎢ (2𝑛 − 2)! 𝑛=1 ⎦ ⎣

⎡

∞

𝑛⎢

∞

=

∑

(−1)

𝑛=0

𝗱 ⎡ 𝖿(0) [ 𝗱𝘅 𝖿 ](0) 2𝑛+1 ⎤ 2𝑛 ⎥ 𝑥 + 𝑥 (2𝑛 + 1)! ⎢ (2𝑛)! ⎥ ⎣ ⎦

𝑛+1 ⎢

= −𝖿 (𝑥)

by right hypothesis

✏ Theorem 1.1 (Taylor series for cosine/sine). ∞

𝖼𝗈𝗌(𝑥) =

t h m

𝑥2𝑛 (−1)𝑛 ∑ (2𝑛)!

𝑥2 𝑥4 𝑥6 + − +⋯ 2 4! 6!

∀𝑥∈ℝ

𝑥2𝑛+1 𝑥3 𝑥5 𝑥7 (−1)𝑛 = 𝑥− + − +⋯ ∑ (2𝑛 + 1)! 3! 5! 7! 𝑛=0

∀𝑥∈ℝ

= 1−

𝑛=0 ∞

𝗌𝗂𝗇(𝑥) =

12

12

📘 Rosenlicht (1968), page 157

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1.3. BASIC PROPERTIES

page 5

Daniel J. Greenhoe

✎PROOF: ∞

∞

𝑥2𝑛 𝑥2𝑛+1 𝗱 𝖼𝗈𝗌(𝑥) = 𝖿 (0) (−1) + [ 𝗱𝘅 𝖿 ](0) (−1)𝑛 ∑ ∑ (2𝑛)! (2𝑛 + 1)! 𝑛=0 𝑛=0 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑛

even terms ∞

=1

odd terms ∞

2𝑛

𝑥2𝑛+1 𝑥 +0 (−1)𝑛 (−1)𝑛 ∑ ∑ (2𝑛)! (2𝑛 + 1)!

by 𝖼𝗈𝗌 initial conditions (Definition 1.1 page 3)

𝑛=0

𝑛=0 ∞

=

by Lemma 1.1 page 3

𝑥2𝑛 (−1)𝑛 ∑ (2𝑛)! 𝑛=0

∞

∞

𝑥2𝑛 𝑥2𝑛+1 𝗱 𝗌𝗂𝗇(𝑥) = 𝖿 (0) + [ 𝗱𝘅 𝖿 ](0) (−1)𝑛 (−1)𝑛 ∑ ∑ (2𝑛)! (2𝑛 + 1)! 𝑛=0 𝑛=0 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ even terms

odd terms

∞

=0

∞

𝑥2𝑛 𝑥2𝑛+1 +1 (−1)𝑛 (−1)𝑛 ∑ ∑ (2𝑛)! (2𝑛 + 1)!

𝑛=0 ∞

=

by Lemma 1.1 page 3

by 𝗌𝗂𝗇 initial conditions (Definition 1.2 page 3)

𝑛=0

2𝑛+1

𝑥 (−1)𝑛 ∑ (2𝑛 + 1)! 𝑛=0

✏ Theorem 1.2. 13 t 𝖼𝗈𝗌(0) = 1 𝖼𝗈𝗌(−𝑥) = 𝖼𝗈𝗌(𝑥) h m 𝗌𝗂𝗇(0) = 0 𝗌𝗂𝗇(−𝑥) = −𝗌𝗂𝗇(𝑥)

∀𝑥∈ℝ ∀𝑥∈ℝ

✎PROOF: 𝖼𝗈𝗌(0) = 1 −

𝑥2 𝑥4 𝑥6 + − +⋯ |𝑥=0 2 4! 6!

by Taylor series for cosine

(Theorem 1.1 page 4)

𝑥3 𝑥5 𝑥7 + − +⋯ |𝑥=0 3! 5! 7!

by Taylor series for sine










=1 𝗌𝗂𝗇(0) = 𝑥 − =0 (−𝑥)2 (−𝑥)4 (−𝑥)6 + − +⋯ 2 4! 6! 𝑥2 𝑥4 𝑥6 =1− + − +⋯ 2 4! 6! = 𝖼𝗈𝗌(𝑥)

𝖼𝗈𝗌(−𝑥) = 1 −

3

5

7

(−𝑥) (−𝑥) (−𝑥) + − +⋯ 3! 5! 7! 𝑥3 𝑥5 𝑥7 =− 𝑥− + − +⋯ [ ] 3! 5! 7!

𝗌𝗂𝗇(−𝑥) = (−𝑥) −

= 𝗌𝗂𝗇(𝑥)

✏ Lemma 1.2. 14 l 𝖼𝗈𝗌(1) > 0 𝑥 ∈ (0 ∶ 2) e m 𝖼𝗈𝗌(2) < 0 13 14

⟹ 𝗌𝗂𝗇(𝑥) > 0

📘 Rosenlicht (1968), page 157 📘 Rosenlicht (1968), page 158



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Daniel J. Greenhoe


✎PROOF: 𝑥2 𝑥4 𝑥6 + − +⋯ |𝑥=1 2 4! 6! 1 1 1 =1− + − +⋯ 2 4! 6! >0

𝖼𝗈𝗌(1) = 1 −

𝑥2 𝑥4 𝑥6 + − +⋯ |𝑥=2 2 4! 6! 4 16 64 =1− + − +⋯ 2 24 720 0 ⦆ 3! ) ( 5! 7! ) ( 9! 11! )

𝗌𝗂𝗇(𝑥) > 0

✏ Proposition 1.1. Let 𝜋 be defined as in Definition 1.3 (page 3). p (A). The value 𝜋 exists in ℝ. r p (B). 2 < 𝜋 < 4. ✎PROOF: 𝖼𝗈𝗌(1) > 0

by Lemma 1.2 page 5

𝖼𝗈𝗌(2) < 0

by Lemma 1.2 page 5

𝜋 ⟹1< 0 cases (by induction): (a) Base case 𝘔 = 1: 𝖼𝗈𝗌(𝑥 + 𝜋) = 𝖼𝗈𝗌𝑥𝖼𝗈𝗌𝜋 − 𝗌𝗂𝗇𝑥𝗌𝗂𝗇𝜋 = 𝖼𝗈𝗌𝑥(−1) − 𝗌𝗂𝗇𝑥(0)

by double angle formulas


by 𝖼𝗈𝗌𝜋 = −1, 𝗌𝗂𝗇𝜋 = 0 results

(Proposition 1.4 page 12)



by 𝖼𝗈𝗌𝜋 = −1, 𝗌𝗂𝗇𝜋 = 0 results


1

= (−1) 𝖼𝗈𝗌𝑥 𝗌𝗂𝗇(𝑥 + 𝜋) = 𝗌𝗂𝗇𝑥𝖼𝗈𝗌𝜋 + 𝖼𝗈𝗌𝑥𝗌𝗂𝗇𝜋 = 𝗌𝗂𝗇𝑥(−1) − 𝖼𝗈𝗌𝑥(0) = (−1)1 𝗌𝗂𝗇𝑥 (b) Inductive step…Proof that 𝘔 case ⟹ 𝘔 + 1 case: 𝖼𝗈𝗌(𝑥 + [𝘔 + 1]𝜋) = 𝖼𝗈𝗌([𝑥 + 𝜋] + 𝘔 𝜋) = (−1)𝘔 𝖼𝗈𝗌(𝑥 + 𝜋)

by induction hypothesis (𝘔 case)

= (−1)𝘔 (−1)𝖼𝗈𝗌(𝑥)

by base case (item (2a) page 14)

= (−1)

𝘔 +1

𝖼𝗈𝗌(𝑥)

⟹ 𝘔 + 1 case 𝗌𝗂𝗇(𝑥 + [𝘔 + 1]𝜋) = 𝗌𝗂𝗇([𝑥 + 𝜋] + 𝘔 𝜋) = (−1)𝘔 𝗌𝗂𝗇(𝑥 + 𝜋) 𝘔

= (−1) (−1)𝗌𝗂𝗇(𝑥)

by induction hypothesis (𝘔 case) by base case (item (2a) page 14)

= (−1)𝘔 +1 𝗌𝗂𝗇(𝑥) ⟹ 𝘔 + 1 case CC

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3. Proof for 𝘔 < 0 cases: Let 𝘕 ≜ −𝘔 … ⟹ 𝘕 > 0. 𝖼𝗈𝗌(𝑥 + 𝘔 𝜋) ≜ 𝖼𝗈𝗌(𝑥 − 𝘕 𝜋)

by definition of 𝘕

= 𝖼𝗈𝗌(𝑥)𝖼𝗈𝗌(−𝘕 𝜋) − 𝗌𝗂𝗇(𝑥)𝗌𝗂𝗇(−𝘕 𝜋)


= 𝖼𝗈𝗌(𝑥)𝖼𝗈𝗌(𝘕 𝜋) + 𝗌𝗂𝗇(𝑥)𝗌𝗂𝗇(𝘕 𝜋)

by Theorem 1.2 page 5


= 𝖼𝗈𝗌(𝑥)𝖼𝗈𝗌(0 + 𝘕 𝜋) + 𝗌𝗂𝗇(𝑥)𝗌𝗂𝗇(0 + 𝘕 𝜋) = 𝖼𝗈𝗌(𝑥)(−1)𝘕 𝖼𝗈𝗌(0) + 𝗌𝗂𝗇(𝑥)(−1)𝘕 𝗌𝗂𝗇(0)

by 𝘔 ≥ 0 results

(item (2) page 14)

= (−1)𝘕 𝖼𝗈𝗌(𝑥)

by 𝖼𝗈𝗌(0) = 1, 𝗌𝗂𝗇(0) = 0 results


≜ (−1)

−𝘔

𝖼𝗈𝗌(𝑥)


= (−1)𝘔 𝖼𝗈𝗌(𝑥) 𝗌𝗂𝗇(𝑥 + 𝘔 𝜋) ≜ 𝗌𝗂𝗇(𝑥 − 𝘕 𝜋)


= 𝗌𝗂𝗇(𝑥)𝗌𝗂𝗇(−𝘕 𝜋) − 𝗌𝗂𝗇(𝑥)𝗌𝗂𝗇(−𝘕 𝜋)


= 𝗌𝗂𝗇(𝑥)𝗌𝗂𝗇(𝘕 𝜋) + 𝗌𝗂𝗇(𝑥)𝗌𝗂𝗇(𝘕 𝜋)



= 𝗌𝗂𝗇(𝑥)𝗌𝗂𝗇(0 + 𝘕 𝜋) + 𝗌𝗂𝗇(𝑥)𝗌𝗂𝗇(0 + 𝘕 𝜋) = 𝗌𝗂𝗇(𝑥)(−1)𝘕 𝗌𝗂𝗇(0) + 𝗌𝗂𝗇(𝑥)(−1)𝘕 𝗌𝗂𝗇(0) 𝘕

= (−1) 𝗌𝗂𝗇(𝑥) ≜ (−1)

−𝘔

𝗌𝗂𝗇(𝑥)

by 𝘔 ≥ 0 results

(item (2) page 14)

by 𝗌𝗂𝗇(0) = 1, 𝗌𝗂𝗇(0) = 0 results



= (−1)𝘔 𝗌𝗂𝗇(𝑥) 4. Proofs using complex exponential: 𝑒𝑖(𝑥+𝘔 𝜋) + 𝑒−𝑖(𝑥+𝘔 𝜋) 2 𝑖𝑥 𝑒 + 𝑒−𝑖𝑥 = 𝑒𝑖𝘔 𝜋 ] [ 2

𝖼𝗈𝗌(𝑥 + 𝘔 𝜋) =

𝘔

= (𝑒𝑖𝜋 ) 𝖼𝗈𝗌𝑥 𝘔

= (−1) 𝖼𝗈𝗌𝑥

by Euler formulas

(Corollary 1.2 page 9)

by Proposition 1.4 page 12

𝘔

= (𝑒𝑖𝜋 ) 𝑠𝑖𝑛𝑥 𝘔

= (−1) 𝗌𝗂𝗇𝑥 𝑒𝑖(𝑥+𝘔 𝜋) = 𝑒𝑖𝘔 𝜋 𝑒𝑖𝑥


−𝑖(𝑥+𝘔 𝜋)

−𝑒 2𝑖 𝑖𝑥 𝑒 − 𝑒−𝑖𝑥 = 𝑒𝑖𝘔 𝜋 [ ] 2𝑖

𝗌𝗂𝗇(𝑥 + 𝘔 𝜋) =

𝑒

𝑖(𝑥+𝘔 𝜋)

by Euler formulas

by Euler formulas


by Euler formulas


by Proposition 1.4 page 12 𝑖𝜋 𝘔

= (𝑒 ) (𝑒𝑖𝑥 )

= (−1)𝘔 𝑒𝑖𝑥


✏ Theorem 1.11 (half-angle formulas/squared identities). t (A). 𝖼𝗈𝗌2 𝑥 = 1/2(1 + 𝖼𝗈𝗌2𝑥) ∀𝑥∈ℝ (C). 𝖼𝗈𝗌2 𝑥 + 𝗌𝗂𝗇2 𝑥 = 1 h m (B). 𝗌𝗂𝗇2 𝑥 = 1/2(1 − 𝖼𝗈𝗌2𝑥) ∀𝑥∈ℝ

∀𝑥∈ℝ

✎PROOF: 𝖼𝗈𝗌2 𝑥 ≜ (𝖼𝗈𝗌𝑥)(𝖼𝗈𝗌𝑥) = 1/2𝖼𝗈𝗌(𝑥 − 𝑥) + 1/2𝖼𝗈𝗌(𝑥 + 𝑥) by product identities = 1/2[1 + 𝖼𝗈𝗌(2𝑥)] 𝗌𝗂𝗇2 𝑥 = (𝗌𝗂𝗇𝑥)(𝗌𝗂𝗇𝑥) = 1/2𝖼𝗈𝗌(𝑥 − 𝑥) − 1/2𝖼𝗈𝗌(𝑥 + 𝑥)


by 𝖼𝗈𝗌(0) = 1 result


by product identities


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= 1/2[1 − 𝖼𝗈𝗌(2𝑥)]

by 𝖼𝗈𝗌(0) = 1 result

2

2

𝖼𝗈𝗌 𝑥 + 𝗌𝗂𝗇 𝑥 = 1/2[1 + 𝖼𝗈𝗌(2𝑥)] + 1/2[1 − 𝖼𝗈𝗌(2𝑥)] = 1


by (A) and (B) note: see also Theorem 1.4 page 7

✏

1.6

Planar Geometry

The harmonic functions 𝖼𝗈𝗌(𝑥) and 𝗌𝗂𝗇(𝑥) are orthogonal to each other in the sense +𝜋

⟨𝖼𝗈𝗌(𝑥) | 𝗌𝗂𝗇(𝑥)⟩ =

∫ −𝜋

𝖼𝗈𝗌(𝑥)𝗌𝗂𝗇(𝑥) 𝖽𝑥 +𝜋

+𝜋

1 1 𝗌𝗂𝗇(𝑥 − 𝑥) 𝖽𝑥 + 𝗌𝗂𝗇(𝑥 + 𝑥) 𝖽𝑥 2∫ 2∫ −𝜋 −𝜋 *0 +𝜋 +𝜋 1 1 𝗌𝗂𝗇(2𝑥) 𝖽𝑥 = 𝖽𝑥 + 𝗌𝗂𝗇(0) 2∫ 2∫ −𝜋 −𝜋 +𝜋 1 1 = − ⋅ 𝖼𝗈𝗌(2𝑥)| 𝖼𝗈𝗌(2𝑥) 2 2 −𝜋 1 = − [𝖼𝗈𝗌(2𝜋) − 𝖼𝗈𝗌(−2𝜋)] 4 =0 =


Because 𝖼𝗈𝗌(𝑥) are 𝗌𝗂𝗇(𝑥) are orthogonal, they can be conveniently represented by the 𝑥 and 𝑦 axes in a plane— because perpendicular axes in a plane are also orthogonal. Vectors in the plane can be represented by linear combinations of 𝖼𝗈𝗌𝑥 and 𝗌𝗂𝗇𝑥. Let tan 𝑥 be defined as tan 𝑥 ≜

𝗌𝗂𝗇𝑥 . 𝖼𝗈𝗌𝑥

We can also define a value 𝜃 to represent the angle between such a vector and the 𝑥-axis such that 𝜃 = tan−1 (

𝑦

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𝗌𝗂𝗇𝜃 𝖼𝗈𝗌𝜃 )

𝖼𝗈𝗌𝜃 ≜ 𝗌𝗂𝗇𝜃 ≜ tan 𝜃 ≜


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𝑥 𝑟 𝑦 𝑟 𝑦 𝑥

sec 𝜃 ≜ csc 𝜃 ≜ cot 𝜃 ≜

𝑟 𝑥 𝑟 𝑦 𝑥 𝑦


1.7. THE POWER OF THE EXPONENTIAL

Daniel J. Greenhoe

page 17

1.7 The power of the exponential ❝Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand

it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth.❞ Benjamin Peirce (1809–1880), American Harvard University mathematician after proving 𝑒𝑖𝜋 = −1 in a lecture. 21

❝Young them.❞

man, in mathematics you don't understand things. You just get used to

John von Neumann (1903–1957), Hungarian-American mathematician, as allegedly told to Gary Zukav by Felix T. Smith, Head of Molecular Physics at Stanford Research Institute, about a “physicist friend”. 22

The following corollary presents one of the most amazing relations in all of mathematics. It shows a simple and compact relationship between the transcendental numbers 𝜋 and 𝑒, the imaginary number 𝑖, and the additive and multiplicative identity elements 0 and 1. The fact that there is any relationship at all is somewhat amazing; but for there to be such an elegant one is truly one of the wonders of the world of numbers. Corollary 1.3. 23 c o 𝑒𝑖𝜋 + 1 = 0 r ✎PROOF: 𝑒𝑖𝑥 |𝑥=𝜋 = [𝖼𝗈𝗌𝑥 + 𝑖𝗌𝗂𝗇𝑥]𝑥=𝜋 = −1 + 𝑖 ⋅ 0

by Euler's identity (Theorem 1.5 page 8) by Proposition 1.4 page 12

= −1

✏ There are many transforms available, several of them integral transforms [𝐀𝖿](𝑠) ≜ ∫𝑡 𝖿(𝑠)𝜅(𝑡, 𝑠) 𝖽𝑠 using different kernels 𝜅(𝑡, 𝑠). But of all of them, two of the most often used themselves use an exponential kernel: ➀ The Laplace Transform with kernel 𝜅 (𝑡, 𝑠) ≜ 𝑒𝑠𝑡 ➁ The Fourier Transform with kernel 𝜅 (𝑡, 𝜔) ≜ 𝑒𝑖𝜔𝑡 . 21

quote: 📘 Kasner and Newman (1940), page 104 image: http://www-history.mcs.st-andrews.ac.uk/history/PictDisplay/Peirce_Benjamin.html 📘 Zukav (1980), page 208 22 quote: image: http://en.wikipedia.org/wiki/John_von_Neumann The quote appears in a footnote in Zukav (1980) that reads like this: Dr. Felix Smith, Head of Molecular Physics, Stanford Research Institute, once related to me the true story of a physicist friend who worked at Los Alamos after World War II. Seeking help on a difficult problem, he went to the great Hungarian mathematician, John von Neumann, who was at Los Alamos as a consultant. “Simple,” said von Neumann. “This can be solved by using the method of characteristics.” After the explanation the physicist said, “I'm afraid I don't understand the method of characteristics.” “Young man,” said von Neumann, “in mathematics you don't understand things, you just get used to them.” 23

📘 Euler (1748), 📘 Euler (1988) ⟨chapter 8?⟩, http://www.daviddarling.info/encyclopedia/E/Eulers_ formula.html Tuesday 3rd October, 2017 3:01am UTC Copyright ©2017 Daniel J. Greenhoe


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Of course, the Fourier kernel is just a special case of the Laplace kernel with 𝑠 = 𝑖𝜔 (𝑖𝜔 is a unit circle in 𝑠 if 𝑠 is depicted as a plane with real and imaginary axes). What is so special about exponential kernels? Is it just that they were discovered sooner than other kernels with other transforms? The answer in general is “no”. The exponential has two properties that makes it extremely special: The exponential is an eigenvalue of any linear time invariant (LTI) operator (Theorem 1.12 page 18). The exponential generates a continuous point spectrum for the differential operator. Theorem 1.12. 24 Let 𝐋 be an operator with kernel 𝗁(𝑡, 𝜔) and ̂ ≜ ⟨𝗁(𝑡, 𝜔) | 𝑒𝑠𝑡 ⟩ 𝗁(𝑠) ( LAPLACE TRANSFORM). t h m {

1. 2.

𝐋 is LINEAR and 𝐋 is TIME-INVARIANT }

⟹

⎧ ⎪ 𝑠𝑡 ⎨𝐋𝑒 = ⎪ ⎩

𝑠𝑡 ̂∗ (−𝑠) 𝑒⏟ 𝗁⏟ eigenvalue

eigenvector

⎫ ⎪ ⎬ ⎪ ⎭

✎PROOF:

𝑠𝑡 𝑠𝑢 [𝐋𝑒 ](𝑠) = ⟨𝑒 | 𝗁((𝑡; 𝑢), 𝑠)⟩ = ⟨𝑒𝑠𝑢 | 𝗁((𝑡 − 𝑢), 𝑠)⟩

= ⟨𝑒 =𝑒

𝑠𝑡

𝑠(𝑡−𝑣)

by time-invariance hypothesis

| 𝗁(𝑣, 𝑠)⟩

−𝑠𝑣

⟨𝑒

by linear hypothesis let 𝑣 = 𝑡 − 𝑢 ⟹ 𝑢 = 𝑡 − 𝑣

| 𝗁(𝑣, 𝑠)⟩

= ⟨𝗁(𝑣, 𝑠) | 𝑒

−𝑠𝑣 ∗

by additivity of ⟨△ | ▽⟩

𝑠𝑡

⟩ 𝑒

by conjugate symmetry of ⟨△ | ▽⟩

(−𝑠)𝑣 ∗

= ⟨𝗁(𝑣, 𝑠) | 𝑒 = 𝗁̂∗ (−𝑠) 𝑒𝑠𝑡

𝑠𝑡 ⟩ 𝑒

̂ by definition of 𝗁(𝑠)

✏

24

📘 Mallat (1999), page 2, …page 2 online: http://www.cmap.polytechnique.fr/~mallat/WTintro.pdf

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CHAPTER 2 TRIGONOMETRIC POLYNOMIALS

❝I

turn aside with a shudder of horror from this lamentable plague of functions which have no derivatives.❞ Charles Hermite (1822 – 1901), French mathematician, in an 1893 letter to Stieltjes, in response to the “pathological” everywhere continuous but nowhere differentiable Weierstrass func1 𝑛 𝑛 tions 𝖿(𝑥) = ∑∞ 𝑛=0 𝑎 𝖼𝗈𝗌(𝑏 𝜋𝑥).

2.1 Trigonometric expansion Theorem 2.1 (DeMoivre's Theorem). t 𝑛 h (𝑟𝑒𝑖𝑥 ) = 𝑟𝑛 (𝖼𝗈𝗌𝑛𝑥 + 𝑖𝗌𝗂𝗇𝑛𝑥) ∀𝑟,𝑥∈ℝ m

✎PROOF: 𝑛

𝑖𝑥 𝑛 𝑖𝑛𝑥 (𝑟𝑒 ) = 𝑟 𝑒 = 𝑟𝑛 (𝖼𝗈𝗌𝑛𝑥 + 𝑖𝗌𝗂𝗇𝑛𝑥)

by Euler's identity (Theorem 1.5 page 8)

✏ The cosine with argument 𝑛𝑥 can be expanded as a polynomial in 𝖼𝗈𝗌(𝑥) (next). Theorem 2.2 (trigonometric expansion). 1

2

quote: 📘 Hermite (1893) translation: 📘 Lakatos (1976), page 19 image: http://www-groups.dcs.sx-and.ac.uk/~history/PictDisplay/Hermite.html 2 📘 Rivlin (1974) page 3 ⟨(1.8)⟩

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Daniel J. Greenhoe

CHAPTER 2. TRIGONOMETRIC POLYNOMIALS

𝑛

⌊2⌋ 𝑘

t h m

𝖼𝗈𝗌(𝑛𝑥) =

∑∑

𝑘=0 𝑚=0

(−1)𝑘+𝑚 (

𝑛 𝑘 (𝖼𝗈𝗌𝑥)𝑛−2(𝑘−𝑚) )( 2𝑘 𝑚)

∀𝑛∈𝕎 and 𝑥∈ℝ

(−1)𝑘+𝑚 (

𝑛 𝑘 (𝗌𝗂𝗇𝑥)𝑛−2(𝑘−𝑚) )( 2𝑘 𝑚)

∀𝑛∈𝕎 and 𝑥∈ℝ

𝑛

⌊2⌋ 𝑘

𝗌𝗂𝗇(𝑛𝑥) =

∑∑

𝑘=0 𝑚=0

✎PROOF: 𝖼𝗈𝗌(𝑛𝑥) = ℜ (𝖼𝗈𝗌𝑛𝑥 + 𝑖𝗌𝗂𝗇𝑛𝑥) = ℜ (𝑒𝑖𝑛𝑥 ) 𝑛

= ℜ [(𝑒𝑖𝑥 ) ] = ℜ [(𝖼𝗈𝗌𝑥 + 𝑖𝗌𝗂𝗇𝑥)𝑛 ] 𝑛

=ℜ

𝑛 (𝖼𝗈𝗌𝑥)𝑛−𝑘 (𝑖𝗌𝗂𝗇𝑥)𝑘 ( ) ∑ 𝑘 [ ] 𝑘∈ℤ 𝑛

=ℜ

𝑛 𝑖𝑘 ( )𝖼𝗈𝗌𝑛−𝑘 𝑥𝗌𝗂𝗇𝑘 𝑥 ∑ 𝑘 [𝑘∈ℤ ]

⎡ 𝑛 𝑛 𝑘 𝑘 𝑛−𝑘 𝑛−𝑘 = ℜ⎢ (𝑘)𝖼𝗈𝗌 𝑥𝗌𝗂𝗇 𝑥 + 𝑖 (𝑘)𝖼𝗈𝗌 𝑥𝗌𝗂𝗇 𝑥 ∑ ⎢ ∑ 𝑘∈{1,5,…,𝑛} ⎣𝑘∈{0,4,…,𝑛} −

𝑛 𝑛 𝑘 𝑘 𝑛−𝑘 𝑛−𝑘 (𝑘)𝖼𝗈𝗌 𝑥𝗌𝗂𝗇 𝑥 + −𝑖 (𝑘)𝖼𝗈𝗌 𝑥𝗌𝗂𝗇 𝑥] ∑ 𝑘∈{3,7,…,𝑛} 𝑘∈{2,6,…,𝑛}

=

𝑛 𝑛 𝑘 𝑘 𝑛−𝑘 𝑛−𝑘 (𝑘)𝖼𝗈𝗌 𝑥𝗌𝗂𝗇 𝑥 − (𝑘)𝖼𝗈𝗌 𝑥𝗌𝗂𝗇 𝑥 ∑ 𝑘∈{0,4,…,𝑛} 𝑘∈{2,6,…,𝑛}

=

𝑘 𝑛 𝑛−𝑘 2 𝖼𝗈𝗌 𝑥𝗌𝗂𝗇𝑘 𝑥 (−1) (𝑘) ∑ 𝑘∈{0,2,…,𝑛}

∑

∑

𝑛

⌊2⌋

=

𝑛 (−1)𝑘 𝖼𝗈𝗌𝑛−2𝑘 𝑥𝗌𝗂𝗇2𝑘 𝑥 ∑ (2𝑘) 𝑘=0 𝑛

⌊2⌋

=

𝑛 (−1)𝑘 𝖼𝗈𝗌𝑛−2𝑘 𝑥(1 − 𝖼𝗈𝗌2 𝑥)𝑘 ∑ (2𝑘) 𝑘=0 𝑛

⎡⌊ 2 ⌋ ⎤ 𝑘 ⎡ ⎤ ⎢ 𝑘 𝑛 𝑘 𝑛−2𝑘 ⎥ ⎢ (−1)𝑚 𝖼𝗈𝗌2𝑚 𝑥⎥ = ⎢ ( )(−1) 𝖼𝗈𝗌 𝑥⎥ ∑ (𝑚) ∑ 2𝑘 ⎢ ⎥ ⎢𝑘=0 ⎥ ⎣𝑚=0 ⎦ ⎣ ⎦ 𝑛

⌊2⌋ 𝑘

=

∑∑

(−1)𝑘+𝑚 (

𝑘=0 𝑚=0

𝑛 𝑘 𝖼𝗈𝗌𝑛−2(𝑘−𝑚) 𝑥 2𝑘)(𝑚)

𝜋 2) 𝜋 = 𝖼𝗈𝗌 (𝑛 [𝑥 − ]) 2𝑛

𝗌𝗂𝗇(𝑛𝑥) = 𝖼𝗈𝗌 (𝑛𝑥 −

𝑛

⌊2⌋ 𝑘

=

∑∑

(−1)𝑘+𝑚 (

𝑘=0 𝑚=0

=

$ CC BY-NC-ND


𝑛 𝑘 𝜋 𝖼𝗈𝗌𝑛−2(𝑘−𝑚) (𝑛 [𝑥 − ]) )( ) 2𝑛 2𝑘 𝑚 version 0.50


2.1. TRIGONOMETRIC EXPANSION

Daniel J. Greenhoe

page 21

𝑛

⌊2⌋ 𝑘

=

∑∑

(−1)𝑘+𝑚 (

𝑛 𝑘 𝜋 𝖼𝗈𝗌𝑛−2(𝑘−𝑚) (𝑛𝑥 − ) 2𝑘)(𝑚) 2

(−1)𝑘+𝑚 (

𝑛 𝑘 𝗌𝗂𝗇𝑛−2(𝑘−𝑚) (𝑛𝑥) )( 2𝑘 𝑚)

𝑘=0 𝑚=0 𝑛

⌊2⌋ 𝑘

=

∑∑

𝑘=0 𝑚=0

✏ Example 2.1. 5 3 e 𝖼𝗈𝗌5𝑥 = 16𝖼𝗈𝗌 𝑥 − 20𝖼𝗈𝗌 𝑥 + 5𝖼𝗈𝗌𝑥 x 𝗌𝗂𝗇5𝑥 = 16𝗌𝗂𝗇5 𝑥 − 20𝗌𝗂𝗇3 𝑥 + 5𝗌𝗂𝗇𝑥. ✎PROOF: 1. Proof using DeMoivre's Theorem (Theorem 2.1 page 19): 𝖼𝗈𝗌5𝑥 + 𝑖𝗌𝗂𝗇5𝑥 = 𝑒𝑖5𝑥 5

= (𝑒𝑖𝑥 )

= (𝖼𝗈𝗌𝑥 + 𝑖𝗌𝗂𝗇𝑥)5 5

=

5 [𝖼𝗈𝗌𝑥]5−𝑘 [𝑖𝗌𝗂𝗇𝑥]𝑘 ∑ (𝑘) 𝑘=0

5 5 5 = ( )[𝖼𝗈𝗌𝑥]5−0 [𝑖𝗌𝗂𝗇𝑥]0 + ( )[𝖼𝗈𝗌𝑥]5−1 [𝑖𝗌𝗂𝗇𝑥]1 + ( )[𝖼𝗈𝗌𝑥]5−2 [𝑖𝗌𝗂𝗇𝑥]2 + 0 1 2 5 5 5 5−3 3 5−4 4 5−5 5 (3)[𝖼𝗈𝗌𝑥] [𝑖𝗌𝗂𝗇𝑥] + (4)[𝖼𝗈𝗌𝑥] [𝑖𝗌𝗂𝗇𝑥] + (5)[𝖼𝗈𝗌𝑥] [𝑖𝗌𝗂𝗇𝑥] = 1𝖼𝗈𝗌5 𝑥 + 𝑖5𝖼𝗈𝗌4 𝑥𝗌𝗂𝗇𝑥 − 10𝖼𝗈𝗌3 𝑥𝗌𝗂𝗇2 𝑥 − 𝑖10𝖼𝗈𝗌2 𝑥𝗌𝗂𝗇3 𝑥 + 5𝖼𝗈𝗌𝑥𝗌𝗂𝗇4 𝑥 + 𝑖1𝗌𝗂𝗇5 𝑥 = [𝖼𝗈𝗌5 𝑥 − 10𝖼𝗈𝗌3 𝑥𝗌𝗂𝗇2 𝑥 + 5𝖼𝗈𝗌𝑥𝗌𝗂𝗇4 𝑥] + 𝑖 [5𝖼𝗈𝗌4 𝑥𝗌𝗂𝗇𝑥 − 10𝖼𝗈𝗌2 𝑥𝗌𝗂𝗇3 𝑥 + 𝗌𝗂𝗇5 𝑥] = [𝖼𝗈𝗌5 𝑥 − 10𝖼𝗈𝗌3 𝑥(1 − 𝖼𝗈𝗌2 𝑥) + 5𝖼𝗈𝗌𝑥(1 − 𝖼𝗈𝗌2 𝑥)(1 − 𝖼𝗈𝗌2 𝑥)] + 𝑖 [5(1 − 𝗌𝗂𝗇2 𝑥)(1 − 𝗌𝗂𝗇2 𝑥)𝗌𝗂𝗇𝑥 − 10(1 − 𝗌𝗂𝗇2 𝑥)𝗌𝗂𝗇3 𝑥 + 𝗌𝗂𝗇5 𝑥] = [𝖼𝗈𝗌5 𝑥 − 10(𝖼𝗈𝗌3 𝑥 − 𝖼𝗈𝗌5 𝑥) + 5𝖼𝗈𝗌𝑥(1 − 2𝖼𝗈𝗌2 𝑥 + 𝖼𝗈𝗌4 𝑥)] + 𝑖 [5(1 − 2𝗌𝗂𝗇2 𝑥 + 𝗌𝗂𝗇4 𝑥)𝗌𝗂𝗇𝑥 − 10(𝗌𝗂𝗇3 𝑥 − 𝗌𝗂𝗇5 𝑥) + 𝗌𝗂𝗇5 𝑥] = [𝖼𝗈𝗌5 𝑥 − 10(𝖼𝗈𝗌3 𝑥 − 𝖼𝗈𝗌5 𝑥) + 5(𝖼𝗈𝗌𝑥 − 2𝖼𝗈𝗌3 𝑥 + 𝖼𝗈𝗌5 𝑥)] + 𝑖 [5(𝗌𝗂𝗇𝑥 − 2𝗌𝗂𝗇3 𝑥 + 𝗌𝗂𝗇5 𝑥) − 10(𝗌𝗂𝗇3 𝑥 − 𝗌𝗂𝗇5 𝑥) + 𝗌𝗂𝗇5 𝑥] = [⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 16𝖼𝗈𝗌5 𝑥 − 20𝖼𝗈𝗌3 𝑥 + 5𝖼𝗈𝗌𝑥] + 𝑖[⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 16𝗌𝗂𝗇5 𝑥 − 20𝗌𝗂𝗇3 𝑥 + 5𝗌𝗂𝗇𝑥] 𝖼𝗈𝗌5𝑥

𝗌𝗂𝗇5𝑥

2. Proof using trigonometric expansion (Theorem 2.2 page 19): 5

⌊2⌋ 𝑘

𝖼𝗈𝗌5𝑥 =

𝑛 𝑘 (−1)𝑘+𝑚 ( )( )(𝖼𝗈𝗌𝑥)𝑛−2(𝑘−𝑚) 2𝑘 𝑚 𝑘=0 𝑚=0

∑∑ 2

=

𝑘

𝑛 𝑘 (−1)𝑘+𝑚 ( )( )(𝖼𝗈𝗌𝑥)5−2(𝑘−𝑚) 2𝑘 𝑚 𝑘=0 𝑚=0

∑∑

5 0 5 1 5 1 = (−1)0 ( )( )𝖼𝗈𝗌5 𝑥 + (−1)1 ( )( )𝖼𝗈𝗌3 𝑥 + (−1)2 ( )( )𝖼𝗈𝗌5 𝑥+ 0 0 2 0 2 1 2 2 2 2 5 1 3 5 3 4 5 (−1) ( )( )𝖼𝗈𝗌 𝑥 + (−1) ( )( )𝖼𝗈𝗌 𝑥 + (−1) ( )( )𝖼𝗈𝗌5 𝑥 4 0 4 1 4 2



version 0.50

CC

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page 22

Daniel J. Greenhoe


𝙓 = 𝗌𝗉𝖺𝗇{𝖼𝗈𝗌(2𝜋𝑛𝑥)|𝑛 = 0, 1, 2, 3}

𝟬

scaling subspace

Figure 2.1: Lattice of harmonic cosines {𝖼𝗈𝗌(𝑛𝑥)|𝑛 = 0, 1, 2, …} = +(1)(1)𝖼𝗈𝗌5 𝑥 − (10)(1)𝖼𝗈𝗌3 𝑥 + (10)(1)𝖼𝗈𝗌5 𝑥 + (5)(1)𝖼𝗈𝗌𝑥 − (5)(2)𝖼𝗈𝗌3 𝑥 + (5)(1)𝖼𝗈𝗌5 𝑥 = +(1 + 10 + 5)𝖼𝗈𝗌5 𝑥 + (−10 − 10)𝖼𝗈𝗌3 𝑥 + 5𝖼𝗈𝗌𝑥 = 16𝖼𝗈𝗌5 𝑥 − 20𝖼𝗈𝗌3 𝑥 + 5𝖼𝗈𝗌𝑥

✏ 3

Example 2.2. 𝑛 𝖼𝗈𝗌𝑛𝑥 0 1 2 3

e x

𝖼𝗈𝗌0𝑥 𝖼𝗈𝗌1𝑥 𝖼𝗈𝗌2𝑥 𝖼𝗈𝗌3𝑥

= = = =

polynomial in 𝖼𝗈𝗌𝑥

𝑛 𝖼𝗈𝗌𝑛𝑥

1 𝖼𝗈𝗌1 𝑥 2𝖼𝗈𝗌2 𝑥 − 1 4𝖼𝗈𝗌3 𝑥 − 3𝖼𝗈𝗌𝑥

4 5 6 7

𝖼𝗈𝗌4𝑥 𝖼𝗈𝗌5𝑥 𝖼𝗈𝗌6𝑥 𝖼𝗈𝗌7𝑥

polynomial in 𝖼𝗈𝗌𝑥 = = = =

8𝖼𝗈𝗌4 𝑥 − 8𝖼𝗈𝗌2 𝑥 + 1 16𝖼𝗈𝗌5 𝑥 − 20𝖼𝗈𝗌3 𝑥 + 5𝖼𝗈𝗌𝑥 32𝖼𝗈𝗌6 𝑥 − 48𝖼𝗈𝗌4 𝑥 + 18𝖼𝗈𝗌2 𝑥 − 1 64𝖼𝗈𝗌7 𝑥 − 112𝖼𝗈𝗌5 𝑥 + 56𝖼𝗈𝗌3 𝑥 − 7𝖼𝗈𝗌𝑥

✎PROOF: 2

⌊2⌋ 𝑘

𝖼𝗈𝗌2𝑥 =

∑∑

(−1)𝑘+𝑚 (

𝑘=0 𝑚=0

3 𝑘 (𝖼𝗈𝗌𝑥)2−2(𝑘−𝑚) )( 2𝑘 𝑚)

3 0 3 1 3 1 = (−1)0 ( )( )𝖼𝗈𝗌2 𝑥 + (−1)1 ( )( )𝖼𝗈𝗌0 𝑥 + (−1)2 ( )( )𝖼𝗈𝗌2 𝑥 0 0 2 0 2 1 = +(1)(1)𝖼𝗈𝗌2 𝑥 − (1)(1) + (1)(1)𝖼𝗈𝗌2 𝑥 = 2𝖼𝗈𝗌2 𝑥 − 1 3

⌊2⌋ 𝑘

𝖼𝗈𝗌3𝑥 =

∑∑

(−1)𝑘+𝑚 (

𝑘=0 𝑚=0

3 𝑘 (𝖼𝗈𝗌𝑥)3−2(𝑘−𝑚) )( 2𝑘 𝑚)

3 0 3 1 3 1 = (−1)0 ( )( )𝖼𝗈𝗌3 𝑥 + (−1)1 ( )( )𝖼𝗈𝗌1 𝑥 + (−1)2 ( )( )𝖼𝗈𝗌3 𝑥 0 0 2 0 2 1 3 0 3 1 3 1 3 1 3 = +( )( )𝖼𝗈𝗌 𝑥 − ( )( )𝖼𝗈𝗌 𝑥 + ( )( )𝖼𝗈𝗌 𝑥 0 0 2 0 2 1 3 1 = +(1)(1)𝖼𝗈𝗌 𝑥 − (3)(1)𝖼𝗈𝗌 𝑥 + (3)(1)𝖼𝗈𝗌3 𝑥 = 4𝖼𝗈𝗌3 𝑥 − 3𝖼𝗈𝗌𝑥 4

⌊2⌋ 𝑘

𝖼𝗈𝗌4𝑥 =

∑∑

(−1)𝑘+𝑚 (

𝑘=0 𝑚=0

4 𝑘 (𝖼𝗈𝗌𝑥)4−2(𝑘−𝑚) )( 2𝑘 𝑚)

3

📘 Abramowitz and Stegun (1972), page 795, 📘 Guillemin (1957), page 593 ⟨(21)⟩, 💻 Sloane (2014) ⟨http://oeis. org/A039991⟩, 💻 Sloane (2014) ⟨http://oeis.org/A028297⟩ CC

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version 0.50


2.1. TRIGONOMETRIC EXPANSION 2

=

𝑘

∑∑

𝑘=0 𝑚=0

(−1)𝑘+𝑚 (

page 23

Daniel J. Greenhoe

4 𝑘 (𝖼𝗈𝗌𝑥)4−2(𝑘−𝑚) 2𝑘)(𝑚)

4 0 4 1 = (−1)0+0 ( (𝖼𝗈𝗌𝑥)4−2(0−0) + (−1)1+0 ( (𝖼𝗈𝗌𝑥)4−2(1−0) 2 ⋅ 0)(0) 2 ⋅ 1)(0) 4 1 4 2 + (−1)1+1 ( (𝖼𝗈𝗌𝑥)4−2(1−1) + (−1)2+0 ( (𝖼𝗈𝗌𝑥)4−2(2−0) 2 ⋅ 1)(1) 2 ⋅ 2)(0) 4 2 4 2 + (−1)2+1 ( (𝖼𝗈𝗌𝑥)4−2(2−1) + (−1)2+2 ( (𝖼𝗈𝗌𝑥)4−2(2−2) )( ) )( 2⋅2 1 2 ⋅ 2 2) = (1)(1)𝖼𝗈𝗌4 𝑥 − (6)(1)𝖼𝗈𝗌2 𝑥 + (6)(1)𝖼𝗈𝗌4 𝑥 + (1)(1)𝖼𝗈𝗌0 𝑥 − (1)(2)𝖼𝗈𝗌2 𝑥 + (1)(1)𝖼𝗈𝗌4 𝑥 = 8𝖼𝗈𝗌4 𝑥 − 8𝖼𝗈𝗌2 𝑥 + 1 𝖼𝗈𝗌5𝑥 = 16𝖼𝗈𝗌5 𝑥 − 20𝖼𝗈𝗌3 𝑥 + 5𝖼𝗈𝗌𝑥

see Example 2.1 page 21

6

⌊2⌋ 𝑘

𝖼𝗈𝗌6𝑥 =

∑∑

𝑘=0 𝑚=0

(−1)𝑘+𝑚 (

6 𝑘 (𝖼𝗈𝗌𝑥)6−2(𝑘−𝑚) )( 2𝑘 𝑚)

6 0 6 1 6 1 6 2 = (−1)0 ( )( )𝖼𝗈𝗌6 𝑥 + (−1)1 ( )( )𝖼𝗈𝗌4 𝑥 + (−1)2 ( )( )𝖼𝗈𝗌6 𝑥 + (−1)2 ( )( )𝖼𝗈𝗌2 𝑥+ 0 0 2 0 2 1 4 0 2 2 3 3 3 6 4 4 6 6 3 6 0 4 6 (−1) ( )( )𝖼𝗈𝗌 𝑥 + (−1) ( )( )𝖼𝗈𝗌 𝑥 + (−1) ( )( )𝖼𝗈𝗌 𝑥 + (−1) ( )( )𝖼𝗈𝗌2 𝑥+ 4 1 4 2 6 0 6 1 6 3 6 3 (−1)5 ( )( )𝖼𝗈𝗌4 𝑥 + (−1)6 ( )( )𝖼𝗈𝗌6 𝑥 6 2 6 3 = +(1)(1)𝖼𝗈𝗌6 𝑥 − (15)(1)𝖼𝗈𝗌4 𝑥 + (15)(1)𝖼𝗈𝗌6 𝑥 + (15)(1)𝖼𝗈𝗌2 𝑥 − (15)(2)𝖼𝗈𝗌4 𝑥 + (15)(1)𝖼𝗈𝗌6 𝑥 − (1)(1)𝖼𝗈𝗌0 𝑥 + (1)(3)𝖼𝗈𝗌2 𝑥 − (1)(3)𝖼𝗈𝗌4 𝑥 + (1)(1)𝖼𝗈𝗌6 𝑥 = 32𝖼𝗈𝗌6 𝑥 − 48𝖼𝗈𝗌4 𝑥 + 18𝖼𝗈𝗌2 𝑥 − 1 7

⌊2⌋ 𝑘

𝖼𝗈𝗌7𝑥 =

∑∑

𝑘=0 𝑚=0 3

=

(−1)𝑘+𝑚 (

𝑛 𝑘 (𝖼𝗈𝗌𝑥)𝑛−2(𝑘−𝑚) 2𝑘)(𝑚)

(−1)𝑘+𝑚 (

𝑛 𝑘 (𝖼𝗈𝗌𝑥)7−2(𝑘−𝑚) 2𝑘)(𝑚)

𝑘

∑∑

𝑘=0 𝑚=0

7 0 7 1 7 1 7 2 = (−1)0 ( )( )𝖼𝗈𝗌7 𝑥 + (−1)1 ( )( )𝖼𝗈𝗌5 𝑥 + (−1)2 ( )( )𝖼𝗈𝗌7 𝑥 + (−1)2 ( )( )𝖼𝗈𝗌3 𝑥 0 0 2 0 2 1 4 0 7 2 7 2 7 3 7 3 + (−1)3 ( )( )𝖼𝗈𝗌5 𝑥 + (−1)4 ( )( )𝖼𝗈𝗌7 𝑥 + (−1)3 ( )( )𝖼𝗈𝗌1 𝑥 + (−1)4 ( )( )𝖼𝗈𝗌3 𝑥 4 1 4 2 6 0 6 1 3 3 5 7 5 6 7 7 + (−1) ( )( )𝖼𝗈𝗌 𝑥 + (−1) ( )( )𝖼𝗈𝗌 𝑥 6 2 6 3 7 5 = (1)(1)𝖼𝗈𝗌 𝑥 − (21)(1)𝖼𝗈𝗌 𝑥 + (21)(1)𝖼𝗈𝗌7 𝑥 + (35)(1)𝖼𝗈𝗌3 𝑥 − (35)(2)𝖼𝗈𝗌5 𝑥 + (35)(1)𝖼𝗈𝗌7 𝑥 − (7)(1)𝖼𝗈𝗌1 𝑥 + (7)(3)𝖼𝗈𝗌3 𝑥 − (7)(3)𝖼𝗈𝗌5 𝑥 + (7)(1)𝖼𝗈𝗌7 𝑥 = (1 + 21 + 35 + 7)𝖼𝗈𝗌7 𝑥 − (21 + 70 + 21)𝖼𝗈𝗌5 𝑥 + (35 + 21)𝖼𝗈𝗌3 𝑥 − (7)𝖼𝗈𝗌1 𝑥 = 64𝖼𝗈𝗌7 𝑥 − 112𝖼𝗈𝗌5 𝑥 + 56𝖼𝗈𝗌3 𝑥 − 7𝖼𝗈𝗌𝑥



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CHAPTER 2. TRIGONOMETRIC POLYNOMIALS 𝙓 = 𝗌𝗉𝖺𝗇{𝑇 𝑛 (𝑥)|𝑛 = 0, 1, 2, 3}

scaling subspace

𝟬

Figure 2.2: Lattice of Chebyshev polynomials {𝑇𝑛 (𝑥)|𝑛 = 0, 1, 2, 3} 1

Note: Trigonometric expansion of 𝖼𝗈𝗌(𝑛𝑥) for particular values of 𝑛 can also be performed with the free software package Maxima™ using syntax illustrated to the right:4

2 3 4 5 6

t r i g e x p a n d ( cos ( 2 * x ) ) ; t r i g e x p a n d ( cos ( 3 * x ) ) ; t r i g e x p a n d ( cos ( 4 * x ) ) ; t r i g e x p a n d ( cos ( 5 * x ) ) ; t r i g e x p a n d ( cos ( 6 * x ) ) ; t r i g e x p a n d ( cos ( 7 * x ) ) ;

✏ Definition 2.1. d The 𝑛th Chebyshev polynomial of the first kind is defined as e 𝑇𝑛 (𝑥) ≜ 𝖼𝗈𝗌𝑛𝑥 where 𝖼𝗈𝗌𝑥 ≜ 𝑥 f

Theorem 2.3. 5 Let 𝑇𝑛 (𝑥) be a CHEBYSHEV POLYNOMIAL with 𝑛 ∈ 𝕎. t 𝑛 is EVEN ⟹ 𝑇𝑛 (𝑥) is EVEN. h m 𝑛 is ODD ⟹ 𝑇𝑛 (𝑥) is ODD. Example 2.3. Let 𝑇𝑛 (𝑥) be a Chebyshev polynomial with 𝑛 ∈ 𝕎. 𝑇0 (𝑥) = 1 𝑇4 (𝑥) = 8𝑥4 − 8𝑥2 + 1 𝑇5 (𝑥) = 16𝑥5 − 20𝑥3 + 5𝑥 e 𝑇1 (𝑥) = 𝑥 x 𝑇 (𝑥) = 2𝑥2 − 1 𝑇 (𝑥) = 32𝑥6 − 48𝑥4 + 18𝑥2 − 1 2 6 𝑇3 (𝑥) = 4𝑥3 − 3𝑥 ✎PROOF:

2.2

Proof of these equations follows directly from Example 2.2 (page 22).

✏

Trigonometric reduction

Theorem 2.2 (page 19) showed that 𝖼𝗈𝗌𝑛𝑥 can be expressed as a polynomial in 𝖼𝗈𝗌𝑥. Conversely, Theorem 2.4 (next) shows that a polynomial in 𝖼𝗈𝗌𝑥 can be expressed as a linear combination of ⦅𝖼𝗈𝗌𝑛𝑥⦆𝑛∈ℤ . Theorem 2.4 (trigonometric reduction). 4

📘 maxima, pages 157–158 ⟨10.5 Trigonometric Functions⟩ 📘 Rivlin (1974) page 5 ⟨(1.13)⟩, 📘 Süli and Mayers (2003) page 242 ⟨Lemma 8.2⟩, 📘 Davidson and Donsig (2010) page 222 ⟨exercise 10.7.A(a)⟩ 5

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Daniel J. Greenhoe

page 25

𝙓 = 𝗌𝗉𝖺𝗇{𝖼𝗈𝗌𝑛 (2𝜋𝑥)|𝑛 = 0, 1, 2, 3}

𝟬

scaling subspace

Figure 2.3: Lattice of exponential cosines {𝖼𝗈𝗌𝑛 𝑥|𝑛 = 0, 1, 2, 3} 𝑛

1 𝑛 𝖼𝗈𝗌 𝑥 = 𝑛 𝖼𝗈𝗌[(𝑛 − 2𝑘)𝑥] ( ∑ 2 𝑘=0 𝑘) 𝑛

𝑛

−1 2 ⎧ 1 𝑛 ⎪ 1 𝑛 ⎪ 2𝑛 ( 𝑛 ) + 2𝑛−1 ∑ (𝑘)𝖼𝗈𝗌[(𝑛 − 2𝑘)𝑥] 𝑘=0 2 ⎪ = ⎨ 𝑛 ⌊2⌋ ⎪ 𝑛 ⎪ 1 𝖼𝗈𝗌[(𝑛 − 2𝑘)𝑥] ⎪ 2𝑛−1 ∑ (𝑘) 𝑘=0 ⎩

t h m

for 𝑛 even

for 𝑛 odd

✎PROOF: 𝖼𝗈𝗌𝑛 𝑥 =

𝑒𝑖𝑥 + 𝑒−𝑖𝑥 ( ) 2

𝑛

𝑛

𝑒𝑖𝑥 + 𝑒−𝑖𝑥 =ℜ ) ] [( 2 𝑛

=ℜ

1 𝑛 𝑖(𝑛−𝑘)𝑥 −𝑖𝑘𝑥 𝑒 (𝑘)𝑒 [ 2𝑛 ∑ ] 𝑘=0 𝑛

𝑛 𝑖(𝑛−2𝑘)𝑥 1 𝑒 =ℜ 𝑛 ( ∑ [ 2 𝑘=0 𝑘) ] 𝑛

=ℜ

1 𝑛 (𝑘) (𝖼𝗈𝗌[(𝑛 − 2𝑘)𝑥] + 𝑖𝗌𝗂𝗇[(𝑛 − 2𝑘)𝑥])] [ 2𝑛 ∑ 𝑘=0 𝑛

𝑛

𝑛 1 𝑛 1 =ℜ 𝑛 𝖼𝗈𝗌[(𝑛 − 2𝑘)𝑥] + 𝑖 𝑛 𝗌𝗂𝗇[(𝑛 − 2𝑘)𝑥] ( ) ( ∑ ∑ 2 𝑘=0 𝑘) [ 2 𝑘=0 𝑘 ] 𝑛

=

𝑛 1 (𝑘)𝖼𝗈𝗌[(𝑛 − 2𝑘)𝑥] 2𝑛 ∑ 𝑘=0 𝑛

−1 2 ⎧ 1 𝑛 1 𝑛 ⎪ ⎪ 2𝑛 ( 𝑛 ) + 2𝑛−1 ∑ (𝑘)𝖼𝗈𝗌[(𝑛 − 2𝑘)𝑥] ∶ 𝑘=0 2 ⎪ =⎨ 𝑛 ⌊2⌋ ⎪ 𝑛 ⎪ 1 𝖼𝗈𝗌[(𝑛 − 2𝑘)𝑥] ∶ ( 𝑛−1 ∑ ⎪ 2 𝑘) ⎩ 𝑘=0

𝑛 even

𝑛 odd

✏ Example 2.4.

6

6

📘 Abramowitz and Stegun (1972), page 795, 💻 Sloane (2014) ⟨http://oeis.org/A100257⟩, 💻 Sloane (2014) ⟨http://oeis.org/A008314⟩ Tuesday 3rd October, 2017 3:01am UTC Copyright ©2017 Daniel J. Greenhoe


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𝑛 𝖼𝗈𝗌𝑛 𝑥


𝑛 𝖼𝗈𝗌𝑛 𝑥

trigonometric reduction

𝖼𝗈𝗌4𝑥 + 4𝖼𝗈𝗌2𝑥 + 3 23 𝖼𝗈𝗌5𝑥 + 5𝖼𝗈𝗌3𝑥 + 10𝖼𝗈𝗌𝑥 5 𝖼𝗈𝗌5 𝑥 = 24 𝖼𝗈𝗌6𝑥 + 6𝖼𝗈𝗌4𝑥 + 15𝖼𝗈𝗌2𝑥 + 10 6 𝖼𝗈𝗌6 𝑥 = 25 𝖼𝗈𝗌7𝑥 + 7𝖼𝗈𝗌5𝑥 + 21𝖼𝗈𝗌3𝑥 + 35𝖼𝗈𝗌𝑥 7 𝖼𝗈𝗌7 𝑥 = 26

0 𝖼𝗈𝗌0 𝑥 = 1 e x

trigonometric reduction

4 𝖼𝗈𝗌4 𝑥 =

1 𝖼𝗈𝗌1 𝑥 = 𝖼𝗈𝗌𝑥 𝖼𝗈𝗌2𝑥 + 1 2 𝖼𝗈𝗌3𝑥 + 3𝖼𝗈𝗌𝑥 3 𝖼𝗈𝗌3 𝑥 = 22 2 𝖼𝗈𝗌2 𝑥 =

✎PROOF: 𝑛

1 𝑛 𝖼𝗈𝗌 𝑥 = 𝑛 𝖼𝗈𝗌 ([𝑛 − 2𝑘]𝑥) ( ∑ 2 𝑘=0 𝑘) | 0

𝑛=0

0

=

1 0 𝖼𝗈𝗌[(0 − 2𝑘)𝑥] ( 0 ∑ 2 𝑘=0 𝑘)

0 = ( )𝖼𝗈𝗌[(0 − 2 ⋅ 0)𝑥] 0 =1 𝑛

𝖼𝗈𝗌1 𝑥 =

𝑛 1 (𝑘)𝖼𝗈𝗌 ([𝑛 − 2𝑘]𝑥)| 2𝑛 ∑ 𝑘=0

𝑛=1

1

=

1 1 (𝑘)𝖼𝗈𝗌[(1 − 2𝑘)𝑥] 21 ∑ 𝑘=0

1 1 1 𝖼𝗈𝗌[(1 − 2 ⋅ 0)𝑥] + ( )𝖼𝗈𝗌[(1 − 2 ⋅ 1)𝑥]] 2 [(0) 1 1 = [1𝖼𝗈𝗌𝑥 + 1𝖼𝗈𝗌(−𝑥)] 2 1 = (𝖼𝗈𝗌𝑥 + 𝖼𝗈𝗌𝑥) 2 = 𝖼𝗈𝗌𝑥 =

𝑛


1 𝑛 (𝑘)𝖼𝗈𝗌 ([𝑛 − 2𝑘]𝑥)| 2𝑛 ∑ 𝑘=0

𝑛=2

2

=

2 1 (𝑘)𝖼𝗈𝗌 ([2 − 2𝑘]𝑥) 22 ∑ 𝑘=0

1 2 2 2 𝖼𝗈𝗌 ([2 − 2 ⋅ 0]𝑥) + ( )𝖼𝗈𝗌 ([2 − 2 ⋅ 1]𝑥) + ( )𝖼𝗈𝗌 ([2 − 2 ⋅ 2]𝑥) +] [( ) 2 0 1 2 2 1 = 2 [1𝖼𝗈𝗌 (2𝑥) + 2𝖼𝗈𝗌 (0𝑥) + 1𝖼𝗈𝗌 (−2𝑥)] 2 1 = 2 [𝖼𝗈𝗌 (2𝑥) + 2 + 𝖼𝗈𝗌 (2𝑥)] 2 1 = [𝖼𝗈𝗌 (2𝑥) + 1] 2 =

𝑛

𝑛 1 𝖼𝗈𝗌 ([𝑛 − 2𝑘]𝑥) 𝖼𝗈𝗌 𝑥 = 𝑛 ( ∑ 2 𝑘=0 𝑘) | 3

𝑛=3

3

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1 [1𝖼𝗈𝗌 (3𝑥) + 3𝖼𝗈𝗌 (1𝑥) + 3𝖼𝗈𝗌 (−1𝑥) + 1𝖼𝗈𝗌 (−3𝑥)] 23 1 = 3 [𝖼𝗈𝗌 (3𝑥) + 3𝖼𝗈𝗌 (𝑥) + 3𝖼𝗈𝗌 (𝑥) + 𝖼𝗈𝗌 (3𝑥)] 2 1 = 2 [𝖼𝗈𝗌 (3𝑥) + 3𝖼𝗈𝗌 (𝑥)] 2 =

𝑛


1 𝑛 (𝑘)𝖼𝗈𝗌 ([𝑛 − 2𝑘]𝑥)| 2𝑛 ∑ 𝑘=0

𝑛=4

4

=

1 4 (𝑘)𝖼𝗈𝗌 ([4 − 2𝑘]𝑥) 24 ∑ 𝑘=0

1 [1𝖼𝗈𝗌 (4𝑥) + 4𝖼𝗈𝗌 (2𝑥) + 6𝖼𝗈𝗌 (0𝑥) + 4𝖼𝗈𝗌 (−2𝑥) + 1𝖼𝗈𝗌 (−4𝑥)] 24 1 = 3 [𝖼𝗈𝗌 (4𝑥) + 4𝖼𝗈𝗌 (2𝑥) + 3] 2

=

5


1 25−1

⌊2⌋

5 𝖼𝗈𝗌[(5 − 2𝑘)𝑥] ∑ (𝑘) 𝑘=0

2

=

5 1 𝖼𝗈𝗌[(5 − 2𝑘)𝑥] ( ∑ 16 𝑘=0 𝑘)

1 5 5 5 𝖼𝗈𝗌5𝑥 + ( )𝖼𝗈𝗌3𝑥 + ( )𝖼𝗈𝗌𝑥] 16 [(0) 1 2 1 = [𝖼𝗈𝗌5𝑥 + 5𝖼𝗈𝗌3𝑥 + 10𝖼𝗈𝗌𝑥] 16

=

6 −1 2

6 1 6 1 𝖼𝗈𝗌6 𝑥 = 6 ( 6 ) + 6−1 𝖼𝗈𝗌[(6 − 2𝑘)𝑥] ( ∑ 𝑘) 2 2 2 𝑘=0 2

6 1 6 1 𝖼𝗈𝗌[(6 − 2𝑘)𝑥] = 6( ) + 5 ( ∑ 2 3 2 𝑘=0 𝑘) 1 1 6 6 6 20 + 𝖼𝗈𝗌6𝑥 + ( )𝖼𝗈𝗌4𝑥( )𝖼𝗈𝗌2𝑥] [( ) 64 32 0 1 2 1 = [𝖼𝗈𝗌6𝑥 + 6𝖼𝗈𝗌4𝑥 + 15𝖼𝗈𝗌2𝑥 + 10] 32

=

7


1 27−1

⌊2⌋

7 𝖼𝗈𝗌[(7 − 2𝑘)𝑥] ∑ (𝑘) 𝑘=0

2

=

7 1 (𝑘)𝖼𝗈𝗌[(7 − 2𝑘)𝑥] 64 ∑ 𝑘=0

1 7 7 7 7 𝖼𝗈𝗌7𝑥 + ( )𝖼𝗈𝗌5𝑥 + ( )𝖼𝗈𝗌3𝑥 + ( )𝖼𝗈𝗌𝑥] [( ) 64 0 1 2 3 1 = [𝖼𝗈𝗌7𝑥 + 7𝖼𝗈𝗌5𝑥 + 21𝖼𝗈𝗌3𝑥 + 35𝖼𝗈𝗌𝑥] 64

=

1

Note: Trigonometric reduction of 𝖼𝗈𝗌𝑛 (𝑥) for particular values of 𝑛 can also be performed with the free software package Maxima™ using syntax illustrated to the right:7 7

2 3 4 5 6

t r i g r e d u c e ( ( cos ( x ) ) ^2) ; t r i g r e d u c e ( ( cos ( x ) ) ^3) ; t r i g r e d u c e ( ( cos ( x ) ) ^4) ; t r i g r e d u c e ( ( cos ( x ) ) ^5) ; t r i g r e d u c e ( ( cos ( x ) ) ^6) ; t r i g r e d u c e ( ( cos ( x ) ) ^7) ;

http://maxima.sourceforge.net/docs/manual/en/maxima_15.html 📘 maxima, page 158 ⟨10.5 Trigonometric Functions⟩



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✏

2.3

Spectral Factorization

Theorem 2.5 (Fejér-Riesz spectral factorization).

8

Let [0, ∞) ⊊ ℝ and

𝘕 𝑖𝑥

𝗉 (𝑒 ) ≜ 𝑎 𝑒𝑖𝑛𝑥 ∑ 𝑛

(Laurent trigonometric polynomial order 2𝘕 )

𝑛=−𝘕 𝘕

𝗊 (𝑒𝑖𝑥 ) ≜

∑

𝑏𝑛 𝑒𝑖𝑛𝑥

(standard trigonometric polynomial order 𝘕 )

𝑛=1

t h 𝗉 (𝑒𝑖𝑥 ) ∈ [0, ∞) m

∀𝑥 ∈ [0, 2𝜋]

⟹

∃ ⦅𝑏𝑛 ⦆𝑛∈ℤ such that { 𝗉 (𝑒𝑖𝑥 ) = 𝗊 (𝑒𝑖𝑥 ) 𝗊∗ (𝑒𝑖𝑥 )

∀𝑥 ∈ ℝ

✎PROOF: 1. Proof that 𝑎𝑛 = 𝑎∗−𝑛 (⦅𝑎𝑛 ⦆𝑛∈ℤ is Hermitian symmetric): Let 𝑎𝑛 ≜ 𝑟𝑛 𝑒𝑖𝜙𝑛 , 𝑟𝑛 , 𝜙𝑛 ∈ ℝ. Then 𝘕

𝗉 (𝑒𝑖𝑛𝑥 ) ≜

∑

𝑎𝑛 𝑒𝑖𝑛𝑥

𝑛=−𝘕 𝘕

=

∑

𝑟𝑛 𝑒𝑖𝜙𝑛 𝑒𝑖𝑛𝑥

𝑛=−𝘕 𝘕

=

∑

𝑟𝑛 𝑒𝑖𝑛𝑥+𝜙𝑛

𝑛=−𝘕 𝘕

=

∑

𝘕

𝑟𝑛 𝖼𝗈𝗌(𝑛𝑥 + 𝜙𝑛 ) + 𝑖

𝑛=−𝘕

∑

𝑟𝑛 𝗌𝗂𝗇(𝑛𝑥 + 𝜙𝑛 )

𝑛=−𝘕

𝘕

=

𝘕

𝘕

𝑟 𝖼𝗈𝗌(𝑛𝑥 + 𝜙𝑛 ) + 𝑖 𝑟0 𝗌𝗂𝗇(0𝑥 + 𝜙0 ) + 𝑟 𝗌𝗂𝗇(𝑛𝑥 + 𝜙𝑛 ) + 𝑟 𝗌𝗂𝗇(−𝑛𝑥 + 𝜙−𝑛 ) ∑ 𝑛 ∑ 𝑛 ∑ −𝑛 [ ] 𝑛=−𝘕 𝑛=1 𝑛=1 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ imaginary part must equal 0 because 𝑝(𝑥) ∈ ℝ 𝘕

=

𝘕

𝘕

𝑟 𝖼𝗈𝗌(𝑛𝑥 + 𝜙𝑛 ) + 𝑖 𝑟0 𝗌𝗂𝗇(𝜙0 ) + 𝑟 𝗌𝗂𝗇(𝑛𝑥 + 𝜙𝑛 ) − 𝑟 𝗌𝗂𝗇(𝑛𝑥 − 𝜙−𝑛 ) ∑ 𝑛 ∑ 𝑛 ∑ −𝑛 [ ] 𝑛=−𝘕 𝑛=1 𝑛=1 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⟹ 𝑟𝑛 = 𝑟−𝑛 , 𝜙𝑛 = −𝜙−𝑛

⟹ 𝑎𝑛 = 𝑎∗−𝑛 , 𝑎0 ∈ ℝ

2. Because the coefficients ⦅𝑐𝑛 ⦆𝑛∈ℤ are Hermitian symmetric, the zeros of 𝑃 (𝑧) occur in conjugate recipriis also a zero of 𝑃 (𝑧) (𝑃 ( 𝜎1∗ ) = 0). In the complex 𝑧 plane, this relationship means zeros are reflected across the unit circle such that col pairs. This means that if 𝜎 ∈ ℂ is a zero of 𝑃 (𝑧) (𝑃 (𝜎) = 0), then

1 𝜎∗

1 1 1 1 1 = = = 𝑒𝑖𝜙 ∗ 𝑖𝜙 ∗ −𝑖𝜙 𝜎 𝑟𝑒 𝑟 (𝑟𝑒 ) 8

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Daniel J. Greenhoe ℑ [𝑧]

page 29

𝑧 = 𝑒𝑖𝑥

ℜ [𝑧]

𝜙 𝑟𝑒𝑖𝜙 c

1 𝑖𝜙

c𝑟 𝑒

3. Because the zeros of 𝗉(𝑧) occur in conjugate recipricol pairs, 𝗉 (𝑒𝑖𝑥 ) can be factored: 𝗉 (𝑒𝑖𝑥 ) = 𝗉(𝑧)|𝑧=𝑒𝑖𝑥 𝘕

=𝑧

−𝑁

𝐶

∏

𝘕

(𝑧 − 𝜎𝑛 )

𝑛=1

𝘕

=𝐶

∏

(𝑧 − 𝜎𝑛 )

𝘕

∏

(𝑧 − 𝜎𝑛 )

𝑛=1

∏( 𝑛=1

𝘕

∏

| 1 𝑧−1 𝑧 − ∗ | ∏ ( 𝜎𝑛 )|| 𝑛=1 𝘕

𝑛=1

=𝐶

𝑛=1

| 1 | 𝜎𝑛∗ )||

𝑧=𝑒𝑖𝑥

𝘕

𝑛=1

=𝐶

∏(

𝑧−

1−

| 1 −1 | 𝑧 )| 𝜎𝑛∗ |

𝘕

(𝑧 − 𝜎𝑛 )

𝑧−1 − 𝜎𝑛∗ ) − ∏( ( 𝑛=1

𝑧=𝑒𝑖𝑥

𝑧=𝑒𝑖𝑥

| 1 | 𝜎𝑛∗ )||

𝑧=𝑒𝑖𝑥

∗

𝘕 ⎡ 𝘕 ⎤ ⎤⎡ 𝘕 1 ⎤⎡ 1 = ⎢𝐶 − ∗ ⎥ ⎢ (𝑧 − 𝜎𝑛 )⎥ ⎢ − 𝜎 𝑛 )⎥ ( ∗ ⎢ ∏ ( 𝜎𝑛 )⎥ ⎢∏ ⎥ ⎥ ⎢∏ 𝑧 ⎣ 𝑛=1 ⎦ ⎦ ⎣ 𝑛=1 ⎦ ⎣ 𝑛=1 ∗ 𝘕 𝘕 | ⎡ ⎤⎡ ⎤ 1 | = ⎢𝐶2 (𝑧 − 𝜎𝑛 )⎥ ⎢𝐶2 − 𝜎 𝑛 )⎥⎥ | ⎢ ∏ ⎥ ⎢ ∏ ( 𝑧∗ ⎣ 𝑛=1 ⎦ ⎣ 𝑛=1 ⎦ |𝑧=𝑒𝑖𝑥 1 = 𝗊(𝑧)𝗊∗ ( ∗ )| 𝑧 𝑧=𝑒𝑖𝑥 𝑖𝑥 ∗ = 𝗊 (𝑒 ) 𝗊 (𝑒𝑖𝑥 )

| | | |𝑧=𝑒𝑖𝑥

✏

2.4 Dirichlet Kernel ❝Dirichlet

alone, not I, nor Cauchy, nor Gauss knows what a completely rigorous proof is. Rather we learn it first from him. When Gauss says he has proved something it is clear; when Cauchy says it, one can wager as much pro as con; when Dirichlet says it, it is certain.❞ Carl Gustav Jacob Jacobi (1804–1851), Jewish-German mathematician

9

quote: image:

9

📘 Schubring (2005), page 558 http://en.wikipedia.org/wiki/File:Carl_Jacobi.jpg, public domain



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The Dirichlet Kernel is critical in proving what is not immediately obvious in examining the Fourier Series—that for a broad class of periodic functions, a function can be recovered from (with uniform convergence) its Fourier Series analysis. Definition 2.2. 10 The Dirichlet Kernel 𝖣𝑛 ∈ ℝ𝕎 with period 𝜏 is defined as d 𝑛 2𝜋 e 1 𝖣𝑛 (𝑥) ≜ 𝑒𝑖 𝜏 𝑘𝑥 f 𝜏 ∑ 𝑘=−𝑛

Proposition 2.1.

11

Let 𝖣𝑛 be the DIRICHLET KERNEL with period 𝜏 (Definition 2.2 page 30).

𝗌𝗂𝗇 𝜋 [2𝑛 + 1]𝑥) p 1 (𝜏 r 𝖣𝑛 (𝑥) = 𝜏 p 𝗌𝗂𝗇( 𝜋𝜏 𝑥) ✎PROOF: 𝑛

𝖣𝑛 (𝑥) ≜

2𝜋 1 𝑒𝑖 𝜏 𝑛𝑥 ∑ 𝜏 𝑘=−𝑛

by definition of 𝖣𝑛

2𝑛

=

2𝑛

(Definition 2.2 page 30)

2𝑛

𝑘 2𝜋 2𝜋 2𝜋 1 1 2𝜋𝑛 1 2𝜋 𝑒𝑖 𝜏 (𝑘−𝑛)𝑥 = 𝑒−𝑖 𝜏 𝑥 𝑒𝑖 𝜏 𝑘𝑥 = 𝑒−𝑖 𝜏 𝑛𝑥 𝑒𝑖 𝜏 𝑥 ) ( ∑ ∑ 𝜏∑ 𝜏 𝜏 𝑘=0 𝑘=0 𝑘=0 2𝜋

2𝑛+1

1 − (𝑒𝑖 𝜏 𝑥 ) 1 −𝑖 2𝜋𝑛 = 𝑒 𝜏 𝑥 2𝜋 𝜏 1 − 𝑒𝑖 𝜏 𝑥

by geometric series

2𝜋

𝜋

𝜋

𝜋

𝑒𝑖 𝜏 (2𝑛+1)𝑥 𝑒−𝑖 𝜏 (2𝑛+1)𝑥 − 𝑒𝑖 𝜏 (2𝑛+1)𝑥 1 2𝜋 1 − 𝑒𝑖 𝜏 (2𝑛+1)𝑥 1 −𝑖 2𝜋𝑛 𝑥 𝜏 = 𝑒−𝑖 𝜏 𝑛𝑥 = 𝑒 𝜋 𝜋 𝜋 2𝜋 𝜏 𝜏 ( 𝑒𝑖 𝜏 𝑥 ) 𝑒−𝑖 𝜏 𝑥 − 𝑒𝑖 𝜏 𝑥 1 − 𝑒𝑖 𝜏 𝑥 2𝜋𝑛 1 2𝜋𝑛 = 𝑒−𝑖 𝜏 𝑥 (𝑒𝑖 𝜏 𝑥 ) 𝜏

−2𝑖𝗌𝗂𝗇 [ 𝜋𝜏 (2𝑛 + 1)𝑥] −2𝑖𝗌𝗂𝗇 [ 𝜋𝜏 𝑥]

1 = 𝜏

𝗌𝗂𝗇 [ 𝜋𝜏 (2𝑛 + 1)𝑥] 𝗌𝗂𝗇 [ 𝜋𝜏 𝑥]

✏ Proposition 2.2. 12 Let 𝖣𝑛 be the DIRICHLET KERNEL with period 𝜏 (Definition 2.2 page 30). 𝜏 p r 𝖣𝑛 (𝑥) 𝖽𝑥 = 1 p ∫ 0 ✎PROOF: 𝜏

𝑛

𝜏

2𝜋 1 𝑒𝑖 𝜏 𝑛𝑥 𝖽𝑥 𝖣𝑛 (𝑥) 𝖽𝑥 ≜ ∑ ∫ ∫ 0 0 𝜏 𝑘=−𝑛

𝑛

𝜏

𝑛

2 𝜏 2

by definition of 𝖣𝑛 (Definition 2.2 page 30)

2𝜋 2 1 = 𝑒𝑖 𝜏 𝑛𝑥 𝖽𝑥 ∑ ∫ 𝜏 𝑘=−𝑛 −𝜏

=

1 2𝜋 2𝜋 𝖼𝗈𝗌( 𝑛𝑥) + 𝑖𝗌𝗂𝗇( 𝑛𝑥) 𝖽𝑥 ∑ 𝜏 𝑘=−𝑛 ∫ 𝜏 𝜏 −𝜏 2

10

📘 Katznelson (2004) page 14, 📘 Heil (2011) pages 443–444, 📘 Folland (1992), pages 33–34 📘 Katznelson (2004) page 14, 📘 Heil (2011) page 444, 📘 Folland (1992), page 34 12 📘 Bruckner et al. (1997) pages 620–621 11

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2.4. DIRICHLET KERNEL

3

1 𝜏 (2𝑛 + 1)

=3

11 10 9 8 7 6 5 4 3 2 1

2 1

−3

−2

−1

1

−1 𝜏 = −1 − 2𝑛+1 3

2

𝜏 = + 2𝑛+1

3

−3

−2

20

−2

−1

1

= 11

2

3

1 3

𝑛=5

1 𝜏 (2𝑛 + 1)

= 21 20

15

15

10

10

5

5 1

−5

1 𝜏 (2𝑛 + 1)

−1 −1 −2 −3

𝑛=1

−3

page 31

Daniel J. Greenhoe

2

3

−3

−2

−1

𝑛 = 10

1 𝜏 (2𝑛 + 1)

1

−5

= 21

2

3

𝑛 = 20

Figure 2.4: 𝖣𝑛 function for 𝑁 = 1, 5, 10, 20. 𝖣𝑛 → comb. (See Proposition 2.1 page 30). 𝜏

𝑛

2 2𝜋 1 𝖼𝗈𝗌( 𝑛𝑥) 𝖽𝑥 = ∑ 𝜏 ∫ 𝜏 𝑘=−𝑛 − 𝜏 2

𝜏

2 𝗌𝗂𝗇( 2𝜋 𝑛𝑥) || 𝜏 1 = | 2𝜋 ∑ 𝜏 𝑘=−𝑛 𝑛 || 𝜏 𝜏 −

𝑛

2

2𝜋 𝜏 ⎡ 𝗌𝗂𝗇( 𝜏 𝑛 2 )

𝑛

1 ⎢ = ∑ ⎢ 𝜏 𝑘=−𝑛 ⎣

2𝜋 𝑛 𝜏

𝗌𝗂𝗇(− 2𝜋 𝑛𝜏 ⎤ 𝜏 2) ⎥ − 2𝜋 ⎥ 𝑛 𝜏 ⎦

𝑛

=

𝗌𝗂𝗇(𝜋𝑛) 𝗌𝗂𝗇(𝜋𝑛) 1𝜏 + ∑ 𝜏 2 𝑘=−𝑛 [ 𝜋𝑛 𝜋𝑛 ]

=

1 𝗌𝗂𝗇(𝜋𝑛) 2 2 [ 𝜋𝑛 ]𝑘=0

=1

✏ Proposition 2.3. Let 𝖣𝑛 be the DIRICHLET KERNEL with period 𝜏 (Definition 2.2 page 30). Let 𝑤𝘕 (the “WIDTH” of 𝖣𝑛 (𝑥)) be the distance between the two points where the center pulse of 𝖣𝑛 (𝑥) intersects the 𝑥 axis. 1 p 𝖣𝑛 (0) = (2𝑛 + 1) 𝜏 r 2𝜏 p 𝑤𝑛 = 2𝑛 + 1 Tuesday 3rd October, 2017 3:01am UTC Copyright ©2017 Daniel J. Greenhoe


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✎PROOF: 𝖣𝑛 (0) = 𝖣𝑛 (𝑥)|𝑡=0 𝗌𝗂𝗇 [ 𝜋𝜏 (2𝑛 + 1)𝑥] || | || 𝗌𝗂𝗇 [ 𝜋𝜏 𝑡] 𝑡=0 𝗱 𝜋 𝗌𝗂𝗇 (2𝑛 + 1)𝑥] || 1 𝗱𝘅 [ 𝜏 = | 𝗱 𝜋 𝜏 || 𝗌𝗂𝗇 𝑡 [𝜏 ] 𝗱𝘅 𝑡=0 1 = 𝜏


by l'Hôpital's rule

𝜋 𝖼𝗈𝗌 [ 𝜋𝜏 (2𝑛 + 1)𝑥] || 1 𝜏 (2𝑛 + 1) = | 𝜋 𝜏 || 𝖼𝗈𝗌 [ 𝜋𝜏 𝑡] 𝜏 𝑡=0 𝜋 1 (2𝑛 + 1) 1 = 𝜏 𝜋 𝜏 1 𝜏

1 = (2𝑛 + 1) 𝜏 The center pulse of kernel 𝖣𝑛 (𝑥) intersects the 𝑥 axis at 𝑡=±

𝜏 (2𝑛 + 1)

which implies 𝑤𝑛 =

𝜏 𝜏 2𝜏 + = . 2𝑛 + 1 2𝑛 + 1 (2𝑛 + 1)

✏ Proposition 2.4. 13 Let 𝖣𝑛 be the DIRICHLET KERNEL with period 𝜏 (Definition 2.2 page 30). p r 𝖣𝑛 (𝑥) = 𝖣𝑛 (−𝑥) (𝖣𝑛 is an EVEN function) p ✎PROOF: 1 𝖣𝑛 (𝑥) = 𝜏 1 = 𝜏 1 = 𝜏

𝗌𝗂𝗇 [ 𝜋𝜏 (2𝑛 + 1)𝑥]


𝗌𝗂𝗇 [ 𝜋𝜏 𝑡] −𝗌𝗂𝗇 [− 𝜋𝜏 (2𝑛 + 1)𝑥]

because 𝗌𝗂𝗇𝑥 is an odd function

−𝗌𝗂𝗇 [− 𝜋𝜏 𝑡] 𝗌𝗂𝗇 [ 𝜋𝜏 (2𝑛 + 1)(−𝑥)] 𝗌𝗂𝗇 [ 𝜋𝜏 (−𝑥)]


= 𝖣𝑛 (−𝑥)

✏ 13

📘 Bruckner et al. (1997) pages 620–621

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Daniel J. Greenhoe

2.5 Trigonometric summations Theorem 2.6 (Lagrange trigonometric identities). 𝗌𝗂𝗇([𝘕 − 21 ]𝑥)

𝘕 −1

t h m

1 𝖼𝗈𝗌(𝑛𝑥) = + ∑ 2 𝑛=0

=

2𝗌𝗂𝗇( 12 𝑥) 𝖼𝗈𝗌([𝘕 − 12 ]𝑥)

𝘕 −1

1 1 𝗌𝗂𝗇(𝑛𝑥) = cot ( 𝑥) + ∑ 2 2 𝑛=0

14

=

2𝗌𝗂𝗇( 12 𝑥)

𝗌𝗂𝗇([𝘕 − 12 ]𝑥) + 𝗌𝗂𝗇( 21 𝑥)

𝖼𝗈𝗌([𝘕

2𝗌𝗂𝗇( 12 𝑥) − 12 ]𝑥) + 𝖼𝗈𝗌( 21 𝑥) 2𝗌𝗂𝗇( 12 𝑥)

∀𝑥∈ℝ

∀𝑥∈ℝ

✎PROOF: 𝘕 −1

∑

𝘕 −1


𝑛=0

∑

𝘕 −1

ℜ𝑒𝑖𝑛𝑥 = ℜ

𝑛=0

∑

𝘕 −1

𝑒𝑖𝑛𝑥 = ℜ

𝑛=0

𝑒𝑖𝑥 ∑( )

𝑛

𝑛=0

𝑖𝘕 𝑥

=ℜ

1−𝑒 [ 1 − 𝑒𝑖𝑥 ] 1

=ℜ

by geometric series 1

1

𝑒𝑖 2 𝘕 𝑥

𝑒−𝑖 2 𝘕 𝑥 − 𝑒𝑖 2 𝘕 𝑥

[( 𝑒𝑖 12 𝑥 )( 𝑒−𝑖 12 𝑥 − 𝑒𝑖 21 𝑥 )]

1 1 ⎡ 1 ⎛ −𝑖 2 𝗌𝗂𝗇( 2 𝘕 𝑥) ⎞⎤ 𝑖 (𝘕 −1)𝑥 ⎜ ⎟⎥ = ℜ⎢(𝑒 2 )⎜ 1 1 ⎢ ⎟⎥ 𝗌𝗂𝗇 𝑥 −𝑖 ( 2 ) ⎠⎦ ⎣ ⎝ 2 1 ⎛ 𝗌𝗂𝗇( 2 𝘕 𝑥) ⎞ 1 ⎟ = 𝖼𝗈𝗌( (𝘕 − 1)𝑥)⎜ 2 ⎜ 𝗌𝗂𝗇 1 𝑥 ⎟ (2 ) ⎠ ⎝

=

− 12 𝗌𝗂𝗇(− 12 𝑥) + 21 𝗌𝗂𝗇([𝘕 − 12 ]𝑥)

1 = + 2

𝘕 −1

∑ 𝑛=0

∑


𝗌𝗂𝗇([𝘕 − 12 ]𝑥) 2𝗌𝗂𝗇( 12 𝑥)

𝘕 −1



𝗌𝗂𝗇( 21 𝑥)

𝘕 −1

ℑ𝑒

𝑖𝑛𝑥

=ℑ

𝑛=0

∑

𝘕 −1

𝑒

𝑖𝑛𝑥

=ℑ

𝑛=0

∑(

𝑛

𝑒𝑖𝑥 )

𝑛=0

𝑖𝘕 𝑥

=ℑ

1−𝑒 [ 1 − 𝑒𝑖𝑥 ] 1

=ℑ

𝑒𝑖 2 𝘕 𝑥

by geometric series 1

1

𝑒−𝑖 2 𝘕 𝑥 − 𝑒𝑖 2 𝘕 𝑥

[( 𝑒𝑖 21 𝑥 )( 𝑒−𝑖𝑥/2 − 𝑒𝑖 12 𝑥 )]

1 1 ⎡ ⎛ − 2 𝑖𝗌𝗂𝗇( 2 𝘕 𝑥) ⎞⎤ ⎟⎥ = ℑ⎢(𝑒𝑖(𝘕 −1)𝑥/2 )⎜ ⎢ ⎜ − 1 𝑖𝗌𝗂𝗇 1 𝑥 ⎟⎥ ( 2 ) ⎠⎦ ⎣ ⎝ 2 14

📃 Muniz (1953) page 140 ⟨“Lagrange's Trigonometric Identities”⟩, 📘 Jeffrey and Dai (2008) pages 128–130 ⟨2.4.1.6 Sines, Cosines, and Tagents of Multiple Angles; (14), (13)⟩ Tuesday 3rd October, 2017 3:01am UTC Copyright ©2017 Daniel J. Greenhoe


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1 (𝘕 − 1)𝑥 ⎛⎜ 𝗌𝗂𝗇( 2 𝘕 𝑥) ⎞⎟ = 𝗌𝗂𝗇 ( )⎜ 1 2 ⎟ ⎝ 𝗌𝗂𝗇( 2 𝑥) ⎠

=

1 𝖼𝗈𝗌(− 12 𝑥) 2

− 12 𝖼𝗈𝗌([𝘕 − 12 ]𝑥)


𝗌𝗂𝗇( 12 𝑥)

1 1 = cot ( 𝑥) + 2 2


𝖼𝗈𝗌([𝘕 − 21 ]𝑥) 2𝗌𝗂𝗇( 12 𝑥)

Note that these results (summed with indices from 𝑛 = 0 to 𝑛 = 𝘕 − 1) are compatible with 📃 Muniz (1953) page 140 (summed with indices from 𝑛 = 1 to 𝑛 = 𝘕 ) as demonstrated next: 𝘕 −1

∑

𝘕


𝑛=0

∑

𝖼𝗈𝗌(𝑛𝑥) + [𝖼𝗈𝗌(0𝑥) − 𝖼𝗈𝗌(𝘕 𝑥)]

𝑛=1

𝗌𝗂𝗇([𝘕 + 12 ]𝑥) ⎤ ⎡ 1 ⎥ + [𝖼𝗈𝗌(0𝑥) − 𝖼𝗈𝗌(𝘕 𝑥)] = ⎢− + 1 ⎢ 2 2𝗌𝗂𝗇( 2 𝑥) ⎥⎦ ⎣ 1 = (1 − ) + 2 1 = + 2 1 = + 2

𝗌𝗂𝗇([𝘕 + 12 ]𝑥) − 2𝗌𝗂𝗇( 21 𝑥)𝖼𝗈𝗌(𝘕 𝑥) 2𝗌𝗂𝗇( 12 𝑥)

𝗌𝗂𝗇([𝘕 + 12 ]𝑥) − 2[𝗌𝗂𝗇([ 21 − 𝘕 ]𝑥) + 𝗌𝗂𝗇[( 12 + 𝘕 )𝑥]] 2𝗌𝗂𝗇( 12 𝑥) 𝗌𝗂𝗇( 12 [2𝘕 − 1]𝑥)


⟹ above result

2𝗌𝗂𝗇( 12 𝑥)

𝘕

𝘕 −1

∑

by 📃 Muniz (1953) page 140


𝑛=0

∑

𝗌𝗂𝗇(𝑛𝑥) + [𝗌𝗂𝗇(0𝑥) − 𝗌𝗂𝗇(𝘕 𝑥)]

𝑛=1

1 1 = cot ( 𝑥) − 2 2 1 1 = cot ( 𝑥) − 2 2 1 1 = cot ( 𝑥) − 2 2 1 1 = cot ( 𝑥) + 2 2

𝖼𝗈𝗌([𝘕 + 12 ]𝑥) 2𝗌𝗂𝗇( 12 𝑥)

+ [0 − 𝗌𝗂𝗇(𝘕 𝑥)]

by 📃 Muniz (1953) page 140

𝖼𝗈𝗌([𝘕 + 12 ]𝑥) − 2𝗌𝗂𝗇( 12 𝑥)𝗌𝗂𝗇(𝘕 𝑥) 2𝗌𝗂𝗇( 12 𝑥) 𝖼𝗈𝗌([𝘕 + 12 ]𝑥) − [𝖼𝗈𝗌([ 12 − 𝘕 ]𝑥) − 𝖼𝗈𝗌([ 21 + 𝘕 ]𝑥)] 2𝗌𝗂𝗇( 12 𝑥) 𝖼𝗈𝗌([𝘕 − 12 ]𝑥)

⟹ above result

2𝗌𝗂𝗇( 12 𝑥)

✏ Theorem 2.7. 15

15

📘 Jeffrey and Dai (2008) pages 128–130 ⟨2.4.1.6 Sines, Cosines, and Tagents of Multiple Angles; (16) and (17)⟩

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𝗌𝗂𝗇([𝘕 − 12 ]𝑥) ⎤ 𝖼𝗈𝗌([𝘕 − 21 ]𝑥) ⎤ ⎡ ⎡ 1 1 1 ⎥ − 𝗌𝗂𝗇(𝑦)⎢ cot ⎥ 𝖼𝗈𝗌(𝑛𝑥 + 𝑦) = 𝖼𝗈𝗌(𝑦)⎢ + ( 2 𝑥) + ∑ 1 1 2 2 ⎢ ⎥ ⎢ 𝑛=0 2𝗌𝗂𝗇( 2 𝑥) ⎦ 2𝗌𝗂𝗇( 2 𝑥) ⎥⎦ ⎣ ⎣ 𝘕 −1 𝗌𝗂𝗇([𝘕 − 12 ]𝑥) ⎤ 𝖼𝗈𝗌([𝘕 − 21 ]𝑥) ⎤ ⎡ ⎡ 1 1 1 ⎥ + 𝗌𝗂𝗇(𝑦)⎢ cot ⎥ 𝗌𝗂𝗇(𝑛𝑥 + 𝑦) = 𝖼𝗈𝗌(𝑦)⎢ + ( 2 𝑥) + ∑ 1 1 2 2 ⎢ ⎥ ⎢ 𝑛=0 2𝗌𝗂𝗇( 2 𝑥) ⎦ 2𝗌𝗂𝗇( 2 𝑥) ⎥⎦ ⎣ ⎣

𝘕 −1

t h m

∀𝑥∈ℝ

∀𝑥∈ℝ

✎PROOF: 𝘕 −1

𝘕 −1

𝖼𝗈𝗌(𝑛𝑥 + 𝑦) =

∑ 𝑛=0

∑

[𝖼𝗈𝗌(𝑛𝑥)𝖼𝗈𝗌(𝑦) − 𝗌𝗂𝗇(𝑛𝑥)𝗌𝗂𝗇(𝑦)]





𝑛=0

𝘕 −1

= 𝖼𝗈𝗌(𝑦)

𝘕 −1

𝖼𝗈𝗌(𝑛𝑥) − 𝗌𝗂𝗇(𝑦)

∑ 𝑛=0

𝘕 −1

∑

∑

𝗌𝗂𝗇(𝑛𝑥)

𝑛=0

𝘕 −1

𝗌𝗂𝗇(𝑛𝑥 + 𝑦) =

∑

[𝖼𝗈𝗌(𝑛𝑥)𝖼𝗈𝗌(𝑦) + 𝗌𝗂𝗇(𝑛𝑥)𝗌𝗂𝗇(𝑦)]

𝑛=0

𝑛=0

𝘕 −1

= 𝖼𝗈𝗌(𝑦)

∑

𝘕 −1

𝖼𝗈𝗌(𝑛𝑥) + 𝗌𝗂𝗇(𝑦)

𝑛=0

∑

𝗌𝗂𝗇(𝑛𝑥)

𝑛=0

✏ Corollary 2.1 (Summation around unit circle). 𝘕 −1

𝘕 −1

t h m

𝘕 −1

2𝑛𝘔 𝜋 2𝑛𝘔 𝜋 2𝑛𝘔 𝜋 2𝑛𝘔 𝜋 = 𝗌𝗂𝗇(𝜃 + = 𝖼𝗈𝗌(𝜃 + 𝗌𝗂𝗇(𝜃 + =0 𝖼𝗈𝗌(𝜃 + ) ) ) ∑ ∑ ∑ 𝘕 𝘕 𝘕 𝘕 ) 𝑛=0 𝑛=0 𝑛=0

𝘕 −1

∑ 𝑛=0

𝘕 −1

𝖼𝗈𝗌2 (𝜃 +

2𝑛𝘔 𝜋 2𝑛𝘔 𝜋 𝘕 = 𝗌𝗂𝗇2 (𝜃 + = ) ) ∑ 𝘕 𝘕 2 𝑛=0

∀𝜃∈ℝ ∀𝘔 ∈ℕ ∀𝜃∈ℝ ∀𝘔 ∈ℕ

✎PROOF: 𝘕 −1

∑ 𝑛=0

𝖼𝗈𝗌(𝜃 +

2𝑛𝘔 𝜋 𝘕 )

𝘕 −1

𝘕 −1

2𝑛𝘔 𝜋 2𝑛𝘔 𝜋 = 𝖼𝗈𝗌(𝜃) 𝖼𝗈𝗌( − 𝗌𝗂𝗇(𝜃) 𝗌𝗂𝗇( ) ∑ ∑ 𝘕 𝘕 ) 𝑛=0 𝑛=0


1 2𝘔 𝜋 𝜋 𝖼𝗈𝗌([𝘕 − 12 ] 2𝘔 ⎡ 1 𝗌𝗂𝗇([𝘕 − 2 ] 𝘕 ) ⎤ ⎡1 ⎤ 𝘕 ) 2𝘔 𝜋 1 ⎥ − 𝗌𝗂𝗇(𝜃)⎢ cot ⎥ = 𝖼𝗈𝗌(𝜃)⎢ + + (2 𝘕 ) 1 2𝘔 𝜋 1 2𝘔 𝜋 ⎢2 ⎥ ⎢ 2 2𝗌𝗂𝗇( 2 𝘕 ) ⎦ 2𝗌𝗂𝗇( 2 𝘕 ) ⎥⎦ ⎣ ⎣


𝘔𝜋 𝖼𝗈𝗌( 𝘔𝘕𝜋 − 2𝘔 𝜋 ) ⎤ ⎡ 1 𝗌𝗂𝗇( 𝘕 − 2𝘔 𝜋 ) ⎤ ⎡1 𝘔 𝜋 ⎥ − 𝗌𝗂𝗇(𝜃)⎢ cot ⎥ = 𝖼𝗈𝗌(𝜃)⎢ − ( 𝘕 )− 𝘔𝜋 𝘔𝜋 ⎢2 ⎥ ⎢ ⎥ 2 2𝗌𝗂𝗇( 𝘕 ) ⎦ 2𝗌𝗂𝗇( 𝘕 ) ⎣ ⎣ ⎦ 𝘔𝜋 ⎡ 1 1 𝗌𝗂𝗇( 𝘕 ) ⎤ ⎥ − 𝗌𝗂𝗇(𝜃) 1 cot 𝘔 𝜋 − 1 cot 𝘔 𝜋 = 𝖼𝗈𝗌(𝜃)⎢ − ( 𝘕 ) 2 ( 𝘕 )] [2 ⎢ 2 2 𝗌𝗂𝗇 𝘔 𝜋 ⎥ ( 𝘕 )⎦ ⎣ = 𝖼𝗈𝗌(𝜃)0 − 𝗌𝗂𝗇(𝜃)0

by 2𝜋 periodicity of 𝗌𝗂𝗇(𝜃) and 𝖼𝗈𝗌(𝜃)

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𝘕 −1


𝘕 −1

𝗌𝗂𝗇 𝜃 + ∑ ( 𝑛=0

2𝑛𝘔 𝜋 𝜋 2𝑛𝘔 𝜋 = 𝖼𝗈𝗌(𝜃 − + 𝘕 ) ∑ 2 𝘕 ) 𝑛=0 𝘕 −1

=

∑ 𝑛=0

𝖼𝗈𝗌(𝜙 +


where 𝜙 ≜ 𝜃 −

=0

𝘕 −1

∑ 𝑛=0

𝖼𝗈𝗌(𝜃 +

by shift identities


𝜋 2

by previous result

2𝑛𝘔 𝜋 2𝑛𝘔 𝜋 𝗌𝗂𝗇 𝜃 + 𝘕 ) ( 𝘕 )

𝘕 −1

𝘕 −1

2𝑛𝘔 𝜋 1 2𝑛𝘔 𝜋 1 2𝑛𝘔 𝜋 2𝑛𝘔 𝜋 =− 𝗌𝗂𝗇([𝜃 + ] − [𝜃 + 𝘕 ]) + 2 ∑ 𝗌𝗂𝗇([𝜃 + 𝘕 ] + [𝜃 + 𝘕 ]) by Theorem 1.8 page 11 2∑ 𝘕 𝑛=0 𝑛=0 𝘕 −1

=−

𝘕 −1

1 4𝑛𝘔 𝜋 :+0 1 𝗌𝗂𝗇(0) 𝗌𝗂𝗇 2𝜃 + ∑ ( 2∑ 2 𝘕 ) 𝑛=0 𝑛=0 𝘕 −1

𝘕 −1

4𝑛𝘔 𝜋 1 4𝑛𝘔 𝜋 1 𝗌𝗂𝗇 + 𝖼𝗈𝗌(2𝜃) 𝗌𝗂𝗇 = 𝗌𝗂𝗇(2𝜃) ∑ ( 𝘕 ) 2 ∑ ( 𝘕 ) 2 𝑛=0 𝑛=0


1 4𝘔 𝜋 𝜋 𝖼𝗈𝗌([𝘕 − 12 ] 4𝘔 ⎤ ⎡ 1 𝗌𝗂𝗇([𝘕 − 2 ] 𝘕 ) ⎤ ⎡1 𝘕 ) 1 4𝘔 𝜋 ⎥ − 𝗌𝗂𝗇(2𝜃)⎢ cot ⎥ + = 𝖼𝗈𝗌(2𝜃)⎢ + (2 𝘕 ) 1 2𝘔 𝜋 1 4𝘔 𝜋 ⎥ ⎥ ⎢2 ⎢2 2𝗌𝗂𝗇 2𝗌𝗂𝗇 (2 𝘕 ) ⎦ (2 𝘕 ) ⎦ ⎣ ⎣


2𝘔 𝜋 𝜋 − 4𝘔 𝜋 ) ⎤ 𝖼𝗈𝗌( 2𝘔 ⎡1 ⎡ 1 𝗌𝗂𝗇( 𝘕 − 4𝘔 𝜋 ) ⎤ 𝘕 2𝘔 𝜋 ⎥ − 𝗌𝗂𝗇(2𝜃)⎢ cot ⎥ = 𝖼𝗈𝗌(2𝜃)⎢ − − ( 𝘕 ) 2𝘔 𝜋 2𝘔 𝜋 ⎥ ⎢2 ⎥ ⎢2 2𝗌𝗂𝗇 2𝗌𝗂𝗇 ( 𝘕 ) ⎦ ( 𝘕 ) ⎣ ⎦ ⎣ 2𝘔 𝜋 ⎡ 1 1 𝗌𝗂𝗇( 𝘕 ) ⎤ ⎥ − 𝗌𝗂𝗇(𝜃) 1 cot 2𝘔 𝜋 − 1 cot 2𝘔 𝜋 = 𝖼𝗈𝗌(𝜃)⎢ − [2 ( 𝘕 ) 2 ( 𝘕 )] ⎢ 2 2 𝗌𝗂𝗇 2𝘔 𝜋 ⎥ ( ) ⎣ ⎦ 𝘕 = 𝖼𝗈𝗌(𝜃)0 − 𝗌𝗂𝗇(𝜃)0

by 2𝜋 periodicity of 𝗌𝗂𝗇(𝜃) and 𝖼𝗈𝗌(𝜃)

=0

𝘕 −1

𝘕 −1

∑ 𝑛=0

𝖼𝗈𝗌2 (𝜃 +

1 4𝑛𝘔 𝜋 2𝑛𝘔 𝜋 = [1 + 𝖼𝗈𝗌(2𝜃 + 𝘕 )] 𝘕 ) 2∑ 𝑛=0


𝘕 −1

=

4𝑛𝘔 𝜋 4𝑛𝘔 𝜋 1 − 𝗌𝗂𝗇(2𝜃)𝗌𝗂𝗇( 1 + 𝖼𝗈𝗌(2𝜃)𝖼𝗈𝗌( ) [ ∑ 2 𝑛=0 𝘕 𝘕 )]

=

1 1 4𝑛𝘔 𝜋 4𝑛𝘔 𝜋 1 1 + 𝖼𝗈𝗌(2𝜃) − 𝗌𝗂𝗇(2𝜃) 𝖼𝗈𝗌( 𝗌𝗂𝗇( ) ∑ ∑ ∑ 2 𝑛=0 2 𝘕 2 𝘕 ) 𝑛=0 𝑛=0

𝘕 −1

𝘕 −1

𝘕 −1

⎡ 1 𝘕 −1 ⎤ 1 1 =⎢ 1⎥ + 𝖼𝗈𝗌(2𝜃)0 − 𝗌𝗂𝗇(2𝜃)0 ∑ ⎢2 ⎥ 2 2 ⎣ 𝑛=0 ⎦ 𝘕 = 2 CC

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2.6. SUMMABILITY KERNELS 𝘕 −1

∑ 𝑛=0

Daniel J. Greenhoe

page 37

𝘕 −1

𝗌𝗂𝗇2 (𝜃 +

2𝑛𝘔 𝜋 𝜋 2𝑛𝘔 𝜋 = 𝖼𝗈𝗌2 (𝜃 − + 𝘕 ) ∑ 2 𝘕 ) 𝑛=0 𝘕 −1

=

∑ 𝑛=0

=

𝖼𝗈𝗌2 (𝜙 +


𝘕 2

by shift identities (Theorem 1.7 page 10)

where 𝜙 ≜ 𝜃 −

𝜋 2

by previous result

✏

2.6 Summability Kernels Definition 2.3. 16 Let ⦅𝜅𝑛 ⦆𝑛∈ℤ be a sequence of CONTINUOUS 2𝜋 PERIODIC functions. The sequence ⦅𝜅𝑛 ⦆𝑛∈ℤ is a summability kernel if 2𝜋 1 1. 𝜅𝑛 (𝑥) 𝖽𝑥 = 1 ∀𝑛∈ℤ and 2𝜋 ∫ 0 d 2𝜋 e 1 2. ∈ ℝ ∀𝑛∈ℤ and f |𝜅𝑛 (𝑥)| 𝖽𝑥 2𝜋 ∫ 0 2𝜋−𝛿

3.

lim

𝑛→∞ ∫ 𝛿

|𝜅𝑛 (𝑥)| 𝖽𝑥 = 0

∀𝑛∈ℤ, 0 0, ∃𝜀

such that

‖𝖿 (𝑥) − 𝖿 (𝑥 + ℎ)‖ < 𝜀

def

⟺ 𝖿 (𝑥) is continuous 3. Proof that (1,2) ⟹ constant property: (a) Suppose there exists 𝑥, 𝑦 ∈ ℝ such that 𝖿(𝑥) ≠ 𝖿(𝑦). (b) Let ⦅𝑥𝑛 ⦆𝑛∈ℕ be a sequence with limit 𝑥 and ⦅𝑦𝑛 ⦆𝑛∈ℕ a sequence with limit 𝑦 (c) Then 0 < ‖𝖿 (𝑥) − 𝖿 (𝑦)‖

by assumption in item (3a) page 54 by (2) and definition of ⦅𝑥𝑛 ⦆ and ⦅𝑦𝑛 ⦆ in item (3b) page 54

= lim ‖𝖿 (𝑥𝑛 ) − 𝖿 (𝑦𝑛 )‖ 𝑛→∞ = lim ‖𝖿 (2𝑚 𝑥𝑛 ) − 𝖿 (2ℓ 𝑦𝑛 )‖ 𝑛→∞

∀𝑚, ℓ ∈ ℤ by (1)

=0 (d) But this is a contradiction, so 𝖿(𝑥) = 𝖿 (𝑦) for all 𝑥, 𝑦 ∈ ℝ, and 𝖿(𝑥) is constant. CC

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Daniel J. Greenhoe

✏ Note that in Proposition 4.8, it is not possible to remove the continuous constraint outright (next two counterexamples). Counterexample 4.1. Let 𝖿 (𝑥) be a function in ℝℝ . c n t

0 for 𝑥 = 0 { 1 otherwise. Then 𝐃𝖿 (𝑥) ≜ √2𝖿(2𝑥) = √2𝖿(𝑥), but 𝖿(𝑥) is not constant. Let 𝖿 (𝑥) ≜

1

𝖿 (𝑥) 𝑡 b

−2

−1

0

1

2

Counterexample 4.2. Let 𝖿(𝑥) be a function in ℝℝ . Let ℚ be the set of rational numbers and ℝ⧵ℚ the set of irrational numbers. c n t

1 for 𝑥 ∈ ℚ { −1 for 𝑥 ∈ ℝ⧵ℚ. Then 𝐃𝖿 (𝑥) ≜ √2𝖿(2𝑥) = √2𝖿(𝑥), but 𝖿(𝑥) is not constant.

1

Let 𝖿 (𝑥) ≜

𝑡 −2

−1−1

1

2

Proposition 4.9 (Operator norm). Let 𝐓 and 𝐃 be as in Definition 4.2 page 50. Let 𝐓 −1 and 𝐃−1 be as in Proposition 4.2 page 50. Let 𝐓∗ and 𝐃∗ be as in Proposition 4.6 page 53. Let ‖⋅‖ and ⟨△ | ▽⟩ be as in Definition B.1 page 87. Let ⦀⋅⦀ be the operator norm (Definition D.7 page 106) induced by ‖⋅‖. p r ⦀𝐓⦀ = ⦀𝐃⦀ = ⦀𝐓∗ ⦀ = ⦀𝐃∗ ⦀ = ⦀𝐓−1 ⦀ = ⦀𝐃−1 ⦀ = 1 p ✎PROOF:

These results follow directly from the fact that 𝐓 and 𝐃 are unitary Theorem D.20 page 122 and Theorem D.21 page 123.


and from

✏

Theorem 4.1. Let 𝐓 and 𝐃 be as in Definition 4.2 page 50. Let 𝐓 −1 and 𝐃−1 be as in Proposition 4.2 page 50. Let ‖⋅‖ and ⟨△ | ▽⟩ be as in Definition B.1 page 87. 1. ‖𝐓𝖿 ‖ = ‖𝐃𝖿‖ = ‖𝖿‖ ∀𝖿∈𝙇𝟤ℝ ( ISOMETRIC IN LENGTH) 𝟤 = ‖𝐃𝖿 − 𝐃𝗀‖ = ‖𝖿 − 𝗀‖ ∀𝖿,𝗀∈𝙇ℝ ( ISOMETRIC IN DISTANCE) t 2. ‖𝐓𝖿−1− 𝐓𝗀‖−1 −1 −1 𝟤 h 3. ‖𝐓 𝖿 − 𝐓 𝗀‖ = ‖𝐃 𝖿 − 𝐃 𝗀‖ = ‖𝖿 − 𝗀‖ ∀𝖿,𝗀∈𝙇ℝ ( ISOMETRIC IN DISTANCE) m 4. ⟨𝐓𝖿 | 𝐓𝗀⟩ = ⟨𝐃𝖿 | 𝐃𝗀⟩ = ⟨𝖿 | 𝗀⟩ ∀𝖿,𝗀∈𝙇𝟤ℝ ( SURJECTIVE) −1 −1 −1 −1 𝟤 5. ⟨𝐓 𝖿 | 𝐓 𝗀⟩ = ⟨𝐃 𝖿 | 𝐃 𝗀⟩ = ⟨𝖿 | 𝗀⟩ ∀𝖿,𝗀∈𝙇ℝ ( SURJECTIVE) ✎PROOF: These results follow directly from the fact that 𝐓 and 𝐃 are unitary Theorem D.20 page 122 and Theorem D.21 page 123.


and from

✏

Proposition 4.10. Let 𝐓 be as in Definition 4.2 page 50. Let 𝐀∗ be the ADJOINT (Definition D.9 page 110) of an operator 𝐀. Let the property “ SELF ADJOINT” be defined as in Definition D.12 (page 116). ∗ p r 𝐓𝑛 = 𝐓𝑛 The operator 𝐓 𝑛 is SELF-ADJOINT p (∑ ) (∑ ) ( [∑ ] ) 𝑛∈ℤ

𝑛∈ℤ

𝑛∈ℤ

✎PROOF: ∑ ⟨(𝑛∈ℤ

𝐓𝑛 𝖿 (𝑥) | 𝗀(𝑥) = 𝖿(𝑥 − 𝑛) | 𝗀(𝑥) ∑ ) ⟩ ⟨𝑛∈ℤ ⟩ =

∑ ⟨𝑛∈ℤ

𝖿(𝑥 + 𝑛) | 𝗀(𝑥)


⟩

by definition of 𝐓 (Definition 4.2 page 50) by commutative property of addition Trigonometric Systems

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Daniel J. Greenhoe

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∑

CHAPTER 4. TRANSVERSAL OPERATORS

⟨𝖿 (𝑥 + 𝑛) | 𝗀(𝑥)⟩

by additive property of ⟨△ | ▽⟩

⟨𝖿 (𝑢) | 𝗀(𝑢 − 𝑛)⟩

where 𝑢 = 𝑥 + 𝑛

𝑛∈ℤ

=

∑

𝑛∈ℤ

= = =

⟨ ⟨ ⟨

𝖿(𝑢) 𝖿(𝑥) 𝖿(𝑥)

∑ | 𝑛∈ℤ

𝗀(𝑢 − 𝑛)

𝗀(𝑥 − 𝑛)

∑ | 𝑛∈ℤ

by additive property of ⟨△ | ▽⟩

⟩

by change of dummy variable: 𝑡 → 𝑢

⟩

𝐓 𝑛 𝗀(𝑥) ⟩ 𝑛∈ℤ

by definition of 𝐓 (Definition 4.2 page 50)

|∑

∗

⟺ ⟺

∑ (𝑛∈ℤ ∑ (𝑛∈ℤ

𝐓

𝑛

)

𝐓𝑛

)

=

∑ (𝑛∈ℤ

𝐓

𝑛

)

is self-adjoint

by definition of operator adjoint by definition of self-adjoint

(Proposition D.3 page 113)

(Definition D.12 page 116)

✏

4.6

Fourier transform properties

Proposition 4.11. Let 𝐓 and 𝐃 be as in Definition 4.2 page 50. Let 𝐁 be the TWO-SIDED LAPLACE TRANSFORM defined as [𝐁𝖿 ](𝑠) ≜ 1. 2.

p r p

3. 4. 5.

𝐁𝐓 𝑛 𝐁𝐃𝑗 𝐃𝐁 𝐁𝐃−1 𝐁−1 𝐃𝐁𝐃

= = = = =

𝑒−𝑠𝑛 𝐁 𝐃−𝑗 𝐁 𝐁𝐃−1 𝐁−1 𝐃−1 𝐁 = 𝐃 𝐃−1 𝐁𝐃−1 = 𝐁

1 √2𝜋

∫ℝ 𝖿(𝑥)𝑒−𝑠𝑥 𝖽𝑥.

∀𝑛∈ℤ ∀𝑗∈ℤ ∀𝑛∈ℤ ∀𝑛∈ℤ

(𝐃−1 is SIMILAR to 𝐃)

∀𝑛∈ℤ

✎PROOF: 𝐁𝐓𝑛 𝖿 (𝑥) = 𝐁𝖿 (𝑥 − 𝑛) =

1 √2𝜋

∫ ℝ

=

1 √2𝜋

∫ ℝ

= 𝑒−𝑠𝑛

by definition of 𝐓 (Definition 4.2 page 50)

𝖿(𝑥 − 𝑛)𝑒−𝑠𝑥 𝖽𝑥

by definition of 𝐁

𝖿(𝑢)𝑒−𝑠(𝑢+𝑛) 𝖽𝑢

where 𝑢 ≜ 𝑥 − 𝑛

1

[ √2𝜋 ∫ ℝ

𝖿 (𝑢)𝑒−𝑠𝑢 𝖽𝑢 ]

= 𝑒−𝑠𝑛 𝐁𝖿(𝑥)


𝐁𝐃𝑗 𝖿 (𝑥) = 𝐁[2𝑗/2 𝖿 (2𝑗 𝑥)]

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by definition of 𝐃 (Definition 4.2 page 50)

=

1 √2𝜋

𝑗/2 𝑗 −𝑠𝑥 𝖽𝑥 [2 𝖿 (2 𝑥)]𝑒 ∫ ℝ

=

1 √2𝜋

𝑗/2 −𝑠2 2−𝑗 𝖽𝑢 [2 𝖿(𝑢)]𝑒 ∫ ℝ

=

√2 1 2 √2𝜋

−𝑗

−𝑗

∫ ℝ

𝖿(𝑢)𝑒−𝑠2

𝑢

by definition of 𝐁 let 𝑢 ≜ 2𝑗 𝑥 ⟹ 𝑥 = 2−𝑗 𝑢

𝖽𝑢


version 0.50


4.6. FOURIER TRANSFORM PROPERTIES

= 𝐃−1

1

[ √2𝜋 ∫ ℝ

𝖿(𝑢)𝑒−𝑠𝑢 𝖽𝑢 ]

= 𝐃−𝑗 𝐁 𝖿(𝑥) 𝐃𝐁 𝖿 (𝑥) = 𝐃 =

1

[ √2𝜋 ∫ ℝ

√2 ℝ √2𝜋 ∫

=

𝖿(𝑥)𝑒−𝑠𝑥 𝖽𝑥 ]

𝖿(𝑥)𝑒−2𝑠𝑥 𝖽𝑥

by definition of 𝐃 (Definition 4.2 page 50)

−1 −𝑠𝑢 [𝐃 𝖿 ](𝑢) 𝑒 𝖽𝑢 ∫ ℝ

−1

let 𝑢 ≜ 2𝑥 ⟹ 𝑥 = 12 𝑢

by Proposition 4.6 page 53 and Proposition 4.7 page 53

= 𝐁𝐃−1 𝖿(𝑥) −1

by Proposition 4.6 page 53 and Proposition 4.7 page 53


√2

1 √2𝜋

page 57


𝑢 𝖿 ( )𝑒−𝑠𝑢 12 𝖽𝑢 ∫ √2𝜋 ℝ 2 𝑢 √2 = 1 𝖿 𝑒−𝑠𝑢 𝖽𝑢 2 ( 2 )] [ √2𝜋 ∫ ℝ =

Daniel J. Greenhoe


−1

𝐁 𝐃 𝐁 = 𝐁 𝐁𝐃

by previous result

=𝐃

by definition of operator inverse (Definition D.4 page 102)

𝐁𝐃−1 𝐁−1 = 𝐃𝐁𝐁−1

by previous result

=𝐃

by definition of operator inverse (Definition D.4 page 102) −1

𝐃𝐁𝐃 = 𝐃𝐃 𝐁

by previous result

=𝐁

by definition of operator inverse (Definition D.4 page 102)

𝐃−1 𝐁𝐃−1 = 𝐃−1 𝐃𝐁

by previous result by definition of operator inverse (Definition D.4 page 102)

=𝐁

✏ ̃ ̃ Corollary 4.1. Let 𝐓 and 𝐃 be as in Definition 4.2 page 50. Let 𝖿(𝜔) ≜ 𝐅𝖿(𝑥) be the FOURIER TRANSFORM 𝟤 (Definition 3.2 page 41) of some function 𝖿 ∈ 𝙇 (Definition B.1 page 87). ℝ 𝑛 −𝑖𝜔𝑛 ̃ ̃ 1. 𝐅𝐓 = 𝑒 𝐅 ̃ 𝑗 = 𝐃−𝑗 𝐅̃ 2. 𝐅𝐃 c ̃ −1 o 3. 𝐃𝐅̃ = 𝐅𝐃 r 4. ̃ −1 𝐅̃ −1 = 𝐅̃ −1 𝐃−1 𝐅̃ 𝐃 = 𝐅𝐃 ̃ ̃ −1 5. 𝐅̃ = 𝐃𝐅𝐃 = 𝐃−1 𝐅𝐃 ✎PROOF:

These results follow directly from Proposition 4.11 page 56. 𝐅̃ = 𝐁|𝑠=𝑖𝜔

✏ ̃ ̃ Proposition 4.12. Let 𝐓 and 𝐃 be as in Definition 4.2 page 50. Let 𝖿(𝜔) ≜ 𝐅𝖿(𝑥) be the FOURIER TRANS𝟤 FORM (Definition 3.2 page 41) of some function 𝖿 ∈ 𝙇ℝ (Definition B.1 page 87). 𝜔 p ̃ 𝑗 𝐓 𝑛 𝖿 (𝑥) = 1 𝑒−𝑖 2𝑗 𝑛 𝖿 ̃ 𝜔 r 𝐅𝐃 ( 2𝑗 ) p 2𝑗/2 ✎PROOF: ̃ 𝑗 𝐓 𝑛 𝖿 (𝑥) = 𝐃−𝑗 𝐅𝐓 ̃ 𝑛 𝖿 (𝑥) 𝐅𝐃 ̃ = 𝐃−𝑗 𝑒−𝑖𝜔𝑛 𝐅𝖿(𝑥) −𝑗 −𝑖𝜔𝑛

=𝐃 𝑒


by Corollary 4.1 page 57 (3)

̃ 𝖿(𝜔)

−𝑗

= 2−𝑗/2 𝑒−𝑖2

by Corollary 4.1 page 57 (3)

𝜔𝑛

𝖿 (̃ 2−𝑗 𝜔)

by Proposition 4.2 page 50 Trigonometric Systems

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Daniel J. Greenhoe


✏ ̃ ̃ be the FOURIER Proposition 4.13. Let 𝐓 be the translation operator (Definition 4.2 page 50). Let 𝖿(𝜔) ≜ 𝐅𝖿(𝑥) 𝟤 ̆ TRANSFORM (Definition 3.2 page 41) of a function 𝖿 ∈ 𝙇ℝ . Let 𝖺(𝜔) be the DTFT (Definition 6.1 page 69) of a 𝟤 sequence ⦅𝑎𝑛 ⦆𝑛∈ℤ ∈ 𝓵ℝ (Definition C.2 page 91). p 𝟤 , 𝜙(𝑥)∈𝙇𝟤 ̃ r 𝐅̃ ̆ 𝜙(𝜔) 𝑎 𝐓 𝑛 𝜙(𝑥) = 𝖺(𝜔) ∀⦅𝑎𝑛 ⦆∈𝓵ℝ ℝ ∑ 𝑛 p 𝑛∈ℤ

✎PROOF: 𝐅̃

∑

𝑎𝑛 𝐓 𝑛 𝜙(𝑥) =

𝑛∈ℤ

∑

̃ 𝑛 𝜙(𝑥) 𝑎𝑛 𝐅𝐓

∑

̃ 𝑎𝑛 𝑒−𝑖𝜔𝑛 𝐅𝜙(𝑥)

𝑛∈ℤ

=

by Corollary 4.1 page 57

𝑛∈ℤ

=

∑ [𝑛∈ℤ

̃ 𝑎𝑛 𝑒−𝑖𝜔𝑛 𝜙(𝜔) ]

̃ by definition of 𝜙(𝜔)

̃ ̆ 𝜙(𝜔) = 𝖺(𝜔)

by definition of DTFT


✏ Theorem 4.2 (Poisson Summation Formula—PSF). 𝟤 3.2 page 41) of a function 𝖿(𝑥) ∈ 𝙇 . ℝ

7

̃ Let 𝖿(𝜔) be the FOURIER TRANSFORM (Definition

2𝜋 √2𝜋 2𝜋 ̂ −1 ̃ ̃ 2𝜋 𝑛 𝑒𝑖 𝜏 𝑛𝑥 𝐅 𝐒𝐅[𝖿(𝑥)] = 𝖿( 𝐓𝜏𝑛 𝖿 (𝑥) = 𝖿(𝑥 + 𝑛𝜏) = √ ∑ ∑ ∑ 𝜏 𝜏 𝑛∈ℤ 𝜏 ) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑛∈ℤ 𝑛∈ℤ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

t h m

operator notation

summation in “time”

𝟤 where 𝐒 ∈ 𝓵ℝ

𝙇𝟤ℝ

summation in “frequency”

is the SAMPLING OPERATOR defined as

𝑛 [𝐒𝖿(𝑥)](𝑛) ≜ 𝖿 ( 2𝜋 𝜏 )

∀𝖿 ∈ 𝙇𝟤(ℝ,ℬ,𝜇) , 𝜏 ∈ ℝ+

✎PROOF: 1. lemma: If 𝗁(𝑥) ≜

∑

̂ Proof: 𝖿(𝑥 + 𝑛𝜏) then 𝗁 ≡ 𝐅̂ −1 𝐅𝗁.

𝑛∈ℤ

Note that 𝗁(𝑥) is periodic with period 𝜏. Because 𝗁 is periodic, it is in the domain of 𝐅̂ and thus 𝗁 ≡ ̂ 𝐅̂ −1 𝐅𝗁. 2. Proof of PSF (this theorem—Theorem 4.2): ∑

𝖿 (𝑥 + 𝑛𝜏) = 𝐅̂ −1 𝐅̂

𝑛∈ℤ

∑

𝖿 (𝑥 + 𝑛𝜏)

by item (1) page 58

𝑛∈ℤ

= 𝐅̂ −1

1 [ √𝜏

𝜏

2𝜋

𝖿 (𝑥 + 𝑛𝜏) 𝑒−𝑖 𝜏 𝑘𝑥 𝖽𝑥 ∑ ∫ ] ) 0 (𝑛∈ℤ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

by definition of 𝐅̂ (Definition 5.1 page 65)

𝐅̂ [∑𝑛∈ℤ 𝖿 (𝑥 + 𝑛𝜏)]

= 𝐅̂ −1

1 [ √𝜏

𝜏

∑∫ 0

2𝜋

𝖿(𝑥 + 𝑛𝜏)𝑒−𝑖 𝜏

𝑛∈ℤ

𝑘𝑥

𝖽𝑥 ]

7

📘 Andrews et al. (2001), page 624, 📘 Knapp (2005b) page 389, 📘 Lasser (1996), page 254, 📘 Rudin (1987), pages 194–195, 📘 Folland (1992), page 337 CC

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4.6. FOURIER TRANSFORM PROPERTIES

= 𝐅̂ −1 = 𝐅̂ −1 =√

1 √ [ 𝜏 1 [ √𝜏

𝑢=(𝑛+1)𝜏

∑∫ 𝑢=𝑛𝜏

2𝜋

𝖿 (𝑢)𝑒−𝑖 𝜏

𝑘(𝑢−𝑛𝜏)

𝑛∈ℤ

∑

:1 𝑒𝑖2𝜋𝑘𝑛

𝑢=(𝑛+1)𝜏

∫ 𝑢=𝑛𝜏

𝑛∈ℤ

page 59

Daniel J. Greenhoe

2𝜋

𝖿(𝑢)𝑒−𝑖 𝜏

𝖽𝑢 ] 𝑘𝑢

where 𝑢 ≜ 𝑥 + 𝑛𝜏 ⟹ 𝑥 = 𝑢 − 𝑛𝜏

𝖽𝑢 ]

−𝑖 2𝜋 𝑘 𝑢 2𝜋 ̂ −1 1 𝐅 𝖿 (𝑢)𝑒 ( 𝜏 ) 𝖽𝑢 [ ] √2𝜋 ∫ 𝜏 𝑢∈ℝ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟


2𝜋 [𝐅𝖿̃ ]( 𝜏 𝑘)

=√

2𝜋 ̂ −1 ̃ 2𝜋 𝐅 [[𝐅𝖿(𝑥)]( 𝑘)] 𝜏 𝜏

by definition of 𝐅̃ (page 41)

=√

2𝜋 ̂ −1 ̃ 𝐅 𝐒𝐅𝖿 𝜏

by definition of 𝐒

=

2𝜋 √2𝜋 ̃ 2𝜋 𝑛 𝑒𝑖 𝜏 𝑛𝑥 𝖿( ) ∑ 𝜏 𝑛∈ℤ 𝜏


✏ ̃ Theorem 4.3 (Inverse Poisson Summation Formula—IPSF). 8 Let 𝖿(𝜔) be the FOURIER TRANSFORM 𝟤 (Definition 3.2 page 41) of a function 𝖿(𝑥) ∈ 𝙇 . ℝ t 𝜏 𝑛 ̃ ̃ 𝜔 − 2𝜋 𝑛 h 𝐓2𝜋/𝜏 = 𝖿(𝑛𝜏)𝑒−𝑖𝜔𝑛𝜏 𝖿 (𝜔) ≜ 𝖿( ) ∑ ∑ ∑ 𝜏 m 𝑛∈ℤ √2𝜋 𝑛∈ℤ 𝑛∈ℤ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ summation in “frequency”

summation in “time”

✎PROOF: 1. lemma: If 𝗁(𝜔) ≜ ∑𝑛∈ℤ 𝖿 ̃ (𝜔 +

2𝜋 𝑛 , 𝜏 )

̂ Proof: then 𝗁 ≡ 𝐅̂ −1 𝐅𝗁.

Note that 𝗁(𝜔) is periodic with period 2𝜋/𝑇 : 𝗁 (𝜔 +

2𝜋 2𝜋 2𝜋 2𝜋 2𝜋 𝖿̃ 𝜔 + 𝖿 ̃ 𝜔 + (𝑛 + 1) ) = 𝖿̃ 𝜔 + ≜ + 𝑛 = 𝑛 ≜ 𝗁(𝜔) ∑ ( ∑ ( ∑ ( 𝜏 ) 𝑛∈ℤ 𝜏 𝜏 ) 𝑛∈ℤ 𝜏 𝜏 ) 𝑛∈ℤ

̂ Because 𝗁 is periodic, it is in the domain of 𝐅̂ and is equivalent to 𝐅̂ −1 𝐅𝗁. 2. Proof of IPSF (this theorem—Theorem 4.3): 2𝜋 𝖿̃ 𝜔 + 𝑛 ∑ ( 𝜏 ) 𝑛∈ℤ 2𝜋 𝑛 𝖿̃ 𝜔 + ∑ ( 𝜏 ) 𝑛∈ℤ

= 𝐅̂ −1 𝐅̂

by item (1) page 59

2𝜋

= 𝐅̂ −1

2𝜋 𝜏 𝜏 ̃ 𝜔 + 2𝜋 𝑛 𝑒−𝑖𝜔 2𝜋/𝜏 𝑘 𝖽𝜔 𝖿( ∑ 𝜏 ) [√ 2𝜋 ∫ ] 0 𝑛∈ℤ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

̃ 𝜔+ 𝐅̂ [∑𝑛∈ℤ 𝖿 (

by definition of 𝐅̂ page 65

2𝜋 𝑛 𝜏 )]

2𝜋

𝜏 𝜏 ̃ 𝜔 + 2𝜋 𝑛 𝑒−𝑖𝜔𝑇 𝑘 𝖽𝜔 =𝐅 𝖿( ∑∫ 𝜏 ) [√ 2𝜋 𝑛∈ℤ ] 0

̂ −1

8

📘 Gauss (1900), page 88



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Daniel J. Greenhoe

= 𝐅̂ −1

𝑢= 2𝜋 (𝑛+1)

𝜏 𝜏 ∑ √ 2𝜋 ∫ [ 2𝜋 𝑛∈ℤ 𝑢= 𝑛

CHAPTER 4. TRANSVERSAL OPERATORS 2𝜋

−𝑖(𝑢− 𝜏 ̃ 𝖿(𝑢)𝑒

𝜏

2𝜋

: 1 𝜏 𝜏 =𝐅 𝑒𝑖2𝜋𝑛𝑘 ∑ ∫2𝜋 𝑛 [√ 2𝜋 𝑛∈ℤ

̂ −1

(𝑛+1)

𝜏

= 𝐅̂ −1

𝑛)𝑇 𝑘

𝖽𝑢 ]

where 𝑢 ≜ 𝜔 +

2𝜋 2𝜋 𝑛 ⟹ 𝜔=𝑢− 𝑛 𝜏 𝜏

−𝑖𝑢𝜏𝑘 ̃ 𝖿 (𝑢)𝑒 𝖽𝑢 ]

𝜏 −𝑖𝑢𝜏𝑘 ̃ 𝖿(𝑢)𝑒 𝖽𝑢 [√ 2𝜋 ∫ ] ℝ

= √𝜏 𝐅̂ −1

1 𝑖𝑢(−𝜏𝑘) ̃ 𝖿(𝑢)𝑒 𝖽𝑢 [ ] √2𝜋 ∫ ℝ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ [𝐅̃

−1

𝖿 ]̃ (−𝑘𝜏)

= √𝜏 𝐅̂ −1 [[𝐅̃ −1 𝖿 ]̃ (−𝑘𝜏)]

by value of 𝐅̃ −1 (Theorem 3.1 page 42)

= √𝜏 𝐅̂ −1 𝐒𝐅̃ −1 𝖿 ̃


= √𝜏 𝐅̂ −1 𝐒𝖿 (𝑥)

by definition of 𝐅̃ (Definition 3.2 page 41

= √𝜏 𝐅̂ −1 𝖿 (−𝑘𝜏) = √𝜏 = =

𝜏

1 2𝜋 ∑ √ 𝜏 𝑘∈ℤ

2𝜋 ∑ √ 𝜏 𝑘∈ℤ

𝜏

√2𝜋

∑


𝖿(−𝑘𝜏)𝑒

𝑖2𝜋

1 2𝜋 𝜏

𝑘𝜔

by definition of 𝐅̂ −1 (Theorem 5.1 page 66)

𝖿(−𝑘𝜏)𝑒𝑖𝑘𝜏𝜔

by definition of 𝐅̂ −1 (Theorem 5.1 page 66)

𝖿(𝑚𝜏)𝑒−𝑖𝜔𝑚𝜏

let 𝑚 ≜ −𝑘

𝑚∈ℤ

✏ Remark 4.1. The left hand side of the Poisson Summation Formula (Theorem 4.2 page 58) is very similar to the Zak Transform 𝐙: 9 (𝐙𝖿)(𝑡, 𝜔) ≜ 𝖿(𝑥 + 𝑛𝜏)𝑒𝑖2𝜋𝑛𝜔 ∑ 𝑛∈ℤ

Remark 4.2. A generalization of the Poisson Summation Formula Trace Formula.10

4.7


is the Selberg

Examples

Example 4.1 (linear functions). 11 Let 𝐓 be the translation operator (Definition 4.2 page 50). Let ℒ(ℂ, ℂ) be the set of all linear functions in 𝙇𝟤ℝ . e 1. {𝑥, 𝐓𝑥} is a basis for ℒ(ℂ, ℂ) and x 2. 𝖿(𝑥) = 𝖿 (1)𝑥 − 𝖿(0)𝐓𝑥 ∀𝖿 ∈ ℒ(ℂ, ℂ) ✎PROOF:

By left hypothesis, 𝖿 is linear; so let 𝖿(𝑥) ≜ 𝑎𝑥 + 𝑏

𝖿(1)𝑥 − 𝖿(0)𝐓𝑥 = 𝖿(1)𝑥 − 𝖿(0)(𝑥 − 1)

by Definition 4.2 page 50

= (𝑎𝑥 + 𝑏)|𝑥=1 𝑥 − (𝑎𝑥 + 𝑏)|𝑥=0 (𝑥 − 1)

by left hypothesis and definition of 𝖿

= (𝑎 + 𝑏)𝑥 − 𝑏(𝑥 − 1) 9

📘 Janssen (1988), page 24, 📘 Zayed (1996), page 482 📘 Lax (2002), page 349, 📘 Selberg (1956), 📘 Terras (1999) 11 📘 Higgins (1996) page 2

10

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Daniel J. Greenhoe

page 61

= 𝑎𝑥 + 𝑏𝑥 − 𝑏𝑥 + 𝑏 = 𝑎𝑥 + 𝑏 = 𝖿 (𝑥)

by left hypothesis and definition of 𝖿

✏ Example 4.2 (Cardinal Series). Let 𝐓 be the translation operator (Definition 4.2 page 50). The Paley-Wiener class of functions 𝙋𝙒𝜎2 are those functions which are “bandlimited” with respect to their Fourier transform (Definition 3.2 page 41). The cardinal series forms an orthogonal basis for such a space. The Fourier coefficients for a projection of a function 𝖿 onto the Cardinal series basis elements is particularly simple—these coefficients are samples of 𝖿(𝑥) taken at regular intervals. In fact, one could represent the coefficients using inner product notation with the Dirac delta distribution 𝛿 as follows: ⟨𝖿(𝑥) | 𝐓 𝑛 𝛿(𝑥)⟩ ≜ 𝖿(𝑥)𝛿(𝑥 − 𝑛) 𝖽𝑥 ≜ 𝖿(𝑛) ∫ ℝ 𝗌𝗂𝗇(𝜋𝑥) 1. 𝐓𝑛 𝑛∈ℕ is a basis for 𝙋𝙒𝜎2 and { } 𝜋𝑥 | ∞ e 𝗌𝗂𝗇(𝜋𝑥) ∀𝖿 ∈ 𝙋𝙒𝜎2 , 𝜎 ≤ 21 x 2. 𝖿(𝑥) = ∑ 𝖿(𝑛)𝐓 𝑛 𝜋𝑥 𝑛=1 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ Cardinal series

Example 4.3 (Fourier Series). 𝑖𝑥 1. {𝐃𝑛 𝑒 | 𝑛∈ℤ} is a basis for 𝙇(0 ∶ 2𝜋) 1 2. 𝖿(𝑥) = 𝛼 𝐃 𝑒𝑖𝑥 ∀𝑥∈(0∶2𝜋), 𝖿∈𝙇(0∶2𝜋) √2𝜋 ∑ 𝑛 𝑛 e 𝑛∈ℤ x 2𝜋 3.

✎PROOF:

𝛼𝑛 ≜

1 √2𝜋

𝖿 (𝑥)𝐃𝑛 𝑒−𝑖𝑥 𝖽𝑥

∫ 0

and where

∀𝖿∈𝙇(0∶2𝜋)

✏

See Theorem 5.1 page 66. 12

Example 4.4 (Fourier Transform). 𝑖𝑥 𝟤 1. {𝐃𝜔 𝑒 |𝜔∈ℝ} is a basis for 𝙇 ℝ e x

2.

𝖿 (𝑥) =

1 √2𝜋

3.

̃ 𝖿(𝜔) ≜

1 √2𝜋

∫ ℝ ∫ ℝ

𝑖𝜔 ̃ 𝖿(𝜔)𝐃 𝑥 𝑒 𝖽𝜔

∀𝖿∈𝙇𝟤ℝ

𝖿(𝑥)𝐃𝜔 𝑒−𝑖𝑥 𝖽𝑥

∀𝖿∈𝙇𝟤ℝ

Example 4.5 (Gabor Transform). 1.

e x

−𝜋𝑥2

{(𝐓𝜏 𝑒

and where

13 𝟤

𝑖𝑥

)(𝐃𝜔 𝑒 )| 𝜏,𝜔∈ℝ} is a basis for 𝙇ℝ

2.

𝖿 (𝑥) =

3.

𝖦 (𝜏, 𝜔) ≜

∫ ℝ

𝖦 (𝜏, 𝜔) 𝐃𝑥 𝑒𝑖𝜔 𝖽𝜔

∀𝑥∈ℝ, 𝖿∈𝙇𝟤ℝ

2

∫ ℝ

and

𝖿 (𝑥)(𝐓𝜏 𝑒−𝜋𝑥 )(𝐃𝜔 𝑒−𝑖𝑥 ) 𝖽𝑥

where

∀𝑥∈ℝ, 𝖿∈𝙇𝟤ℝ

Example 4.6 (wavelets). Let 𝜓(𝑥) be a wavelet. 𝑘 𝑛 𝟤 and 1. {𝐃 𝐓 𝜓(𝑥)| 𝑘,𝑛∈ℤ} is a basis for 𝙇 ℝ 𝑘 𝑛 𝟤 2. 𝖿(𝑥) = 𝛼 𝐃 𝐓 𝜓(𝑥) ∀𝖿∈𝙇ℝ where e ∑ ∑ 𝑘,𝑛 𝑘∈ℤ 𝑛∈ℤ x 3. 𝛼𝑛 ≜ 𝖿(𝑥)𝐃𝑘 𝐓𝑛 𝜓 ∗ (𝑥) 𝖽𝑥 ∀𝖿∈𝙇𝟤ℝ ∫ ℝ 12 13

cross reference: Definition 3.2 page 41 📃 Gabor (1946), 📘 Qian and Chen (1996) ⟨Chapter 3⟩, 📘 Forster and Massopust (2009) page 32 ⟨Definition 1.69⟩



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4.8

Daniel J. Greenhoe


Cardinal Series and Sampling

4.8.1

Cardinal series basis

The Paley-Wiener class of functions (next definition) are those with a bandlimited Fourier transform. The cardinal series forms an orthogonal basis for such a space (Theorem 4.5 page 62). In a frame ⦅𝒙𝑛 ⦆𝑛∈ℤ with frame operator 𝐒 on a Hilbert Space 𝙃 with inner product ⟨△ | ▽⟩, a function 𝖿(𝑥) in the space spanned by the frame can be represented by 𝖿(𝑥) = 𝖿 | 𝐒−1 𝒙𝑛 ⟩ 𝒙𝑛 . ⏟⏟⏟⏟⏟⏟⏟ ∑ ⟨ 𝑛∈ℤ

“Fourier coefficient”

If the frame is orthonormal (giving an orthonormal basis), then 𝐒 = 𝐒−1 = 𝐈 and 𝖿(𝑥) = ⟨𝖿 | 𝒙𝑛 ⟩ 𝒙𝑛 . ∑ 𝑛∈ℤ

In the case of the cardinal series, the Fourier coefficients are particularly simple—these coefficients are samples of 𝖿 taken at regular intervals (Theorem 4.6 page 63). In fact, one could represent the coefficients using inner product notation with the Dirac delta distribution 𝛿 as follows: ⟨𝖿(𝑥) | 𝛿(𝑥 − 𝑛𝜏)⟩ ≜

∫ ℝ

𝖿(𝑥)𝛿(𝑥 − 𝑛𝜏) 𝖽𝑡 ≜ 𝖿(𝑛𝜏)

Definition 4.5. 14 𝑝 A function 𝖿 ∈ ℂℂ is in the Paley-Wiener class of functions 𝙋𝙒𝜎 if 𝙇𝑝 (−𝜎 ∶ 𝜎) such that d there exists 𝖥 ∈ 𝜎 e 𝖿 (𝑥) = 𝖥(𝜔)𝑒𝑖𝑥𝜔 𝖽𝜔 (𝖿 has a BANDLIMITED Fourier transform 𝖥 with bandwidth 𝜎 ) f ∫ −𝜎 for 𝑝 ∈ [1 ∶ ∞) and 𝜎 ∈ (0 ∶ ∞). Theorem 4.4 (Paley-Wiener Theorem for Functions). 15 Let 𝖿 be an ENTIRE FUNCTION (the domain of 𝖿 is the entire complex plane ℂ). Let 𝜎 ∈ ℝ+ . + t 1. ∃𝐶 ∈ ℝ such that |𝖿(𝑧)| ≤ 𝐶𝑒𝜎|𝑧| ( EXPONENTIAL TYPE) and h {𝖿 ∈ 𝙋𝙒𝜎2 } ⟺ 𝟤 { 2. 𝖿 ∈ 𝙇ℝ } m

1

−3𝜏

−2𝜏

−𝜏

𝜏

2𝜏

3𝜏

Theorem 4.5 (Cardinal sequence). t 1 h { ≥ 2𝜎 } ⟹ The sequence 𝜏 m

16

𝗌𝗂𝗇 [ 𝜋𝜏 (𝑥 − 𝑛𝜏)] ⦅

𝜋 (𝑥 − 𝑛𝜏) 𝜏

⦆

is an ORTHONORMAL BASIS for 𝙋𝙒𝜎2 . 𝑛∈ℤ

14

📘 Higgins (1996) page 52 ⟨Definition 6.15⟩ 📘 Boas (1954) page 103 ⟨6.8.1 Theorem of Paley and Wiener⟩, 📘 Katznelson (2004) page 212 ⟨7.4 Theorem⟩, 📘 Zygmund (2002) pages 272–273 ⟨(7⋅2) THEOREM OF PALEY-WIENER⟩, 📘 YOSIDA (1980) PAGE 161, 📘 RUDIN (1987) PAGE 375 ⟨19.3 THEOREM⟩, 📘 YOUNG (2001) PAGE 85 ⟨THEOREM 18⟩ 16 📘 Higgins (1996) page 52 ⟨Definition 6.15⟩, 📘 Hardy (1941) ⟨orthonormality⟩, 📘 Higgins (1985), page 56 ⟨H1.; historical notes⟩ 15

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4.8. CARDINAL SERIES AND SAMPLING

Theorem 4.6 (Sampling Theorem). t h m {

1. 2.

𝖿 ∈ 𝙋𝙒𝜎2 1 ≥ 2𝜎 𝜏

17

𝗌𝗂𝗇[ 𝜋𝜏 (𝑥 − 𝑛𝜏)]

∞

and

⟹

}

page 63

Daniel J. Greenhoe

𝖿(𝑥) =

. 𝖿(𝑛𝜏) 𝜋 ∑ (𝑥 − 𝑛𝜏) 𝑛=1 𝜏 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ CARDINAL SERIES

✎PROOF: Let

𝗌(𝑥) ≜

𝗌𝗂𝗇 [ 𝜋𝜏 𝑥] 𝜋 𝑥 𝜏

̃ ⟺ 𝗌(𝜔) =

1 𝜏 ∶ |𝑓 | ≤ 2𝜏 { 0 ∶ otherwise

1. Proof that the set is orthonormal: see 📘 Hardy (1941) 2. Proof that the set is a basis: 𝖿 (𝑥) =

∫ 𝜔

𝑖𝜔𝑡 ̃ 𝖿 (𝜔)𝑒 𝖽𝜔

by inverse Fourier transform (Theorem 3.1 page 42)

𝑖𝜔𝑡 ̃ 𝑇 𝖿d̃ (𝜔)𝗌(𝜔)𝑒 𝖽𝜔

if 𝑊 ≤

= 𝑇 𝖿𝖽 (𝑥) ⋆ 𝗌(𝑥)

1 2𝑇 by Convolution theorem (Theorem 3.6 page 44)

=𝑇

by convolution definition (Definition 3.3 page 44)

=

∫ 𝜔

=𝑇 =𝑇

[𝖿𝖽 (𝑢)]𝗌(𝑥 − 𝑢) 𝖽𝑢 ∫ 𝑢 ∑ ∫ 𝑢[ 𝑛 ∑∫ 𝑢

𝖿(𝑢)𝛿(𝑢 − 𝑛𝜏) 𝗌(𝑥 − 𝑢) 𝖽𝑢 ]

by sampling definition (Theorem 4.7 page 63)

𝖿(𝑢)𝗌(𝑥 − 𝑢)𝛿(𝑢 − 𝑛𝜏) 𝖽𝑢

𝑛∈ℤ

=𝑇

∑

𝖿(𝑛𝜏)𝗌(𝑥 − 𝑛𝜏)

by property of Dirac delta distribution

𝑛∈ℤ

=𝑇

∑

𝑛∈ℤ

𝖿(𝑛𝜏)

𝗌𝗂𝗇 [ 𝜋𝜏 (𝑥 − 𝑛𝜏)] 𝜋 (𝑥 𝜏

by definition of 𝗌(𝑥)

− 𝑛𝜏)

✏

4.8.2 Sampling ̃ If 𝖿𝖽 (𝑥) is the function 𝖿(𝑥) sampled at rate 1/𝑇 , then 𝖿d̃ (𝜔) is simply 𝖿(𝜔) replicated every 1/𝑇 Hertz and scaled by 1/𝑇 . This is proven in Theorem 4.7 (next) and illustrated in Figure 4.1 (page 64). Theorem 4.7. Let 𝖿, 𝖿𝖽 ∈ 𝙇𝟤ℝ and 𝖿,̃ 𝖿d̃ ∈ 𝙇𝟤ℝ be their respective fourier transforms. Let 𝖿𝖽 (𝑥) be the sampled 𝖿(𝑥) such that 𝖿𝖽 (𝑥) ≜ 𝖿(𝑥)𝛿(𝑥 − 𝑛𝜏). ∑ 𝑛∈ℤ

Then t 2𝜋 2𝜋 h 𝖿d̃ (𝜔) = 𝖿 ̃ (𝜔 − 𝑛 . ∑ 𝜏 𝑛∈ℤ 𝜏 ) m 17

📘 Whittaker (1915), 📘 Kotelnikov (1933), 📘 Whittaker (1935), 📃 Shannon (1948) ⟨Theorem 13⟩, 📃 Shannon (1949) page 11 📘 II (1991) page 1, 📘 Nashed and Walter (1991), 📘 Higgins (1996) page 5, 📘 Young (2001) pages 90–91 ⟨THE PALEY-WIENER SPACE⟩ Tuesday 3rd October, 2017 3:01am UTC Copyright ©2017 Daniel J. Greenhoe


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Daniel J. Greenhoe


𝐴

−𝑊 @ @ @

@ @

−2 𝜏

𝐴 𝜏

@ −𝑊

−1 𝜏

̃ @ 𝖿(𝜔) @ @ +𝑊

𝑓

𝖿d̃ (𝜔) at sample rate 𝜏1 = 3𝑊 (“oversampling”) @ @ @ @ @ @ @ @ @ 1 2 +𝑊 = 3𝑊 𝜏

𝐴

𝖿d̃ (𝜔) at sample rate

𝜏

1 𝜏

= 2𝑊 (at Nyquist rate)

Q Q Q Q Q Q 𝜏Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q −3 −2 −1 1 2 3 −𝑊 +𝑊 = 2𝑊 𝜏

𝜏

𝜏

𝜏

𝜏

−5 𝜏

−4 𝜏

−3 𝜏

−2 𝜏

−1 𝜏

1 𝜏

=𝑊

2 𝜏

3 𝜏

𝑓

𝜏

𝖿d̃ (𝜔) at sample rate 𝜏1 = 𝑊 (“undersampling”) 𝐴 PPP PPP PPP PPP PPP PPP P P PPP PPP P𝜏PP PPP PPP PP PP −6 𝜏

𝑓

4 𝜏

5 𝜏

𝑓

6 𝜏

Figure 4.1: Sampling in frequency domain ✎PROOF: 𝖿d̃ (𝜔) ≜ = =

𝖿𝖽 (𝑥)𝑒−𝑖𝜔𝑡 𝖽𝑡 ∫ 𝑡 ∑ ∫ 𝑡 [𝑛∈ℤ

𝖿 (𝑥)𝛿(𝑥 − 𝑛𝜏) 𝑒−𝑖𝜔𝑡 𝖽𝑡 ]

𝖿(𝑥)𝛿(𝑥 − 𝑛𝜏)𝑒−𝑖𝜔𝑡 𝖽𝑡 ∑∫ 𝑡

𝑛∈ℤ

=

∑

𝖿(𝑛𝜏)𝑒−𝑖𝜔𝑛𝜏

by definition of 𝛿

𝑛∈ℤ

=

2𝜋 2𝜋 𝖿 ̃ (𝜔 + 𝑛 ∑ 𝜏 𝑛∈ℤ 𝜏 )

=

2𝜋 2𝜋 𝖿 ̃ (𝜔 − 𝑛 ∑ 𝜏 𝑛∈ℤ 𝜏 )

by IPSF


✏ Suppose a waveform 𝖿(𝑥) is sampled at every time 𝑇 generating a sequence of sampled values 𝖿(𝑛𝜏). Then in general, we can approximate 𝖿(𝑥) by using interpolation between the points 𝖿(𝑛𝜏). Interpolation can be performed using several interpolation techniques. In general all techniques lead only to an approximation of 𝖿(𝑥). However, if 𝖿(𝑥) is bandlimited with 1 bandwidth 𝑊 ≤ 2𝑇 , then 𝖿(𝑥) is perfectly reconstructed (not just approximated) from the sampled values 𝖿(𝑛𝜏) (Theorem 4.6 page 63).

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CHAPTER 5 FOURIER SERIES

…et la nouveauté de l'objet, jointe à son importance, a déterminé la classe à couronner cet ouvrage, en observant cependant que la manière dont l`auteur parvient à ses équations n`est pas exempte de difficultés, et que son analyse, pour les intégrer, laisse encore quelque chose à désirer, soit relativement à la généralité, soit même du coté de la rigueur.❞ ❝

…and the innovation of the subject, together with its importance, convinced the committee to crown this work. By observing however that the way in which the author arrives at his equations is not free from difficulties, and the analysis of which, to integrate them, still leaves something to be desired, either relative to generality, or even on the side of rigour.❞ ❝

A competition awards committee consisting of the mathematical giants Lagrange, Laplace, Legendre, and others, commenting on Fourier's 1807 landmark paper Dissertation on the propagation of heat in solid bodies that introduced the Fourier Series. 1

5.1 Definition The Fourier Series expansion of a periodic function is simply a complex trigonometric polynomial. In the special case that the periodic function is even, then the Fourier Series expansion is a cosine polynomial. Definition 5.1. 2 𝟤 is defined as The Fourier Series operator 𝐅̂ ∶ 𝙇𝟤ℝ → 𝓵ℝ d 𝜏 2𝜋 e −𝑖 𝑛𝑥 1 ∀𝖿 ∈ {𝖿 ∈ 𝙇𝟤ℝ |𝖿 is periodic with period 𝜏 } f [𝐅𝖿̂ ](𝑛) ≜ √𝜏 ∫ 𝖿(𝑥)𝑒 𝜏 𝖽𝑥 0 1

📘 Lagrange et al. (1812b), page 374, 📘 Lagrange et al. (1812a), page 112, 📘 Kahane (2008) page 199 translation: assisted by Google Translate, 📘 Castanedo (2005) ⟨chapter 2 footnote 5⟩ paper: 📘 Fourier (1807) 2 📘 Katznelson (2004) page 3 quote:

page 66

5.2

Daniel J. Greenhoe

CHAPTER 5. FOURIER SERIES

Inverse Fourier Series operator

Theorem 5.1. Let 𝐅̂ be the Fourier Series operator. The inverse Fourier Series operator 𝐅̂ −1 is given by t 𝑖 2𝜋 𝑛𝑥 h 𝟤 ̂ −1 ⦅𝗑̃𝑛 ⦆ ](𝑥) ≜ 1 𝜏 ̃ 𝐅 𝗑 𝑒 ∀⦅𝗑̃𝑛 ⦆∈𝓵ℝ [ 𝑛 𝑛∈ℤ m √𝜏 ∑ 𝑛∈ℤ

✎PROOF:

The proof of the pointwise convergence of the Fourier Series is notoriously difficult. It was conjectured in 1913 by Nokolai Luzin that the Fourier Series for all square summable periodic functions are pointwise convergent: 📃 Luzin (1913)

Fifty-three years later (1966) at a conference in Moscow, Lennart Axel Edvard Carleson presented one of the most spectacular results ever in mathematics; he demonstrated that the Luzin conjecture is indeed correct. Carleson formally published his result that same year: 📃 Carleson (1966) Carleson's proof is expounded upon in Reyna's (2002) 175 page book: 📘

de Reyna (2002)

Interestingly enough, Carleson started out trying to disprove Luzin's conjecture. Carleson said this in an interview published in 2001:3 “…the problem of course presents itself already when you are a student and I was thinking about the problem on and off, but the situation was more interesting than that. The great authority in those days was Zygmund and he was completely convinced that what one should produce was not a proof but a counter-example. When I was a young student in the United States, I met Zygmund and I had an idea how to produce some very complicated functions for a counter-example and Zygmund encouraged me very much to do so. I was thinking about it for about 15 years on and off, on how to make these counter-examples work and the interesting thing that happened was that I realised why there should be a counter-example and how you should produce it. I thought I really understood what was the background and then to my amazement I could prove that this “correct” counter-example couldn't exist and I suddenly realised that what you should try to do was the opposite, you should try to prove what was not fashionable, namely to prove convergence. The most important aspect in solving a mathematical problem is the conviction of what is the true result. Then it took 2 or 3 years using the techniques that had been developed during the past 20 years or so.” For now, if you just want some intuitive justification for the Fourier Series, and you can somehow imagine that the Dirichlet kernel generates a comb function of Dirac delta functions, then perhaps what follows may help (or not). It is certainly not mathematically rigorous and is by no means a real proof (but at least it is less than 175 pages).

𝜏

2𝜋

−𝑖 𝑛𝑥 −1 ̂ −1 1 [𝐅̂ 𝐅𝗑 ] (𝑥) = 𝐅̂ [ √𝜏 ∫ 𝗑(𝑥)𝑒 𝜏 𝖽𝑥] 0 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

=

1 √𝜏

=

1 √𝜏

=

3

1 √𝜏

̂ 𝐅𝗑 𝜏

1

∑ [ √𝜏 ∫ 0 𝑛∈ℤ ∑

1 √𝜏

∑

1 √𝜏

𝑛∈ℤ

𝑛∈ℤ

𝜏

∫ 0 𝜏

∫ 0

2𝜋

𝗑(𝑢)𝑒−𝑖 𝜏 2𝜋

𝗑(𝑢)𝑒−𝑖 𝜏 2𝜋

𝗑(𝑢)𝑒𝑖 𝜏

𝑛𝑢

by definition of 𝐅̂ Definition 5.1 page 65

2𝜋

𝖽𝑢 𝑒𝑖 𝜏 ]

𝑛𝑢 𝑖 2𝜋 𝑛𝑥 𝜏

𝑒

𝑛(𝑥−𝑢)

𝑛𝑥

by definition of 𝐅̂ −1

𝖽𝑢

𝖽𝑢

📘 Carleson and Engquist (2001), http://www.gap-system.org/~history/Biographies/Carleson.html

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5.2. INVERSE FOURIER SERIES OPERATOR 𝜏

=

𝗑(𝑢)

∫ 0

Daniel J. Greenhoe

page 67

2𝜋 1 𝑒𝑖 𝜏 𝑛(𝑥−𝑢) 𝖽𝑢 ∑ 𝜏 𝑛∈ℤ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

lim𝑁→∞ 𝖣𝑛 (𝑥) 𝜏

=

𝗑(𝑢)

∫ 0

𝛿(𝑥 − 𝑢 − 𝑛𝜏)

∑ [𝑛∈ℤ

]

𝖽𝑢

𝑢=𝜏

=

𝗑(𝑢)𝛿(𝑥 − 𝑢 − 𝑛𝜏) 𝖽𝑢

∑∫ 𝑢=0

𝑛∈ℤ

𝑣−𝑛𝜏=𝜏

=

∑∫ 𝑣−𝑛𝜏=0

𝗑(𝑣 − 𝑛𝜏)𝛿(𝑥 − 𝑣) 𝖽𝑣

where 𝑣 ≜ 𝑢 + 𝑛𝜏

𝑛∈ℤ

𝑣=(𝑛+1)𝜏

=

∑∫ 𝑣=𝑛𝜏

𝗑(𝑣 − 𝑛𝜏)𝛿(𝑥 − 𝑣) 𝖽𝑣

where 𝑣 ≜ 𝑢 + 𝑛𝜏

𝗑(𝑣)𝛿(𝑥 − 𝑣) 𝖽𝑣

because 𝗑 is periodic with period 𝜏

𝑛∈ℤ

𝑣=(𝑛+1)𝜏

=

∑∫ 𝑣=𝑛𝜏

𝑛∈ℤ

=

∫ ℝ

𝗑(𝑣)𝛿(𝑥 − 𝑣) 𝖽𝑣

= 𝗑(𝑥) ̃ = 𝐈𝗑(𝑛)

−1 [𝐅̂ 𝐅̂ 𝗑̃] (𝑛) = 𝐅̂

=

by definition of 𝐈 (Definition D.4 page 102)

2𝜋

1 [ √𝜏

1 √𝜏

∑

𝑖𝜏 ̃ 𝗑(𝑘)𝑒

𝑘𝑥

𝑘∈ℤ 𝜏

∫ 0

1 √ [ 𝜏

by definition of 𝐅̂ −1

] 2𝜋

∑

𝑖𝜏 ̃ 𝗑(𝑘)𝑒

𝑘∈ℤ

2𝜋

𝑘𝑥

]

𝑒−𝑖 𝜏

𝑛𝑥

𝖽𝑥

by definition of 𝐅̂ (Definition 5.1 page 65)

𝜏

=

1 𝑖 2𝜋 (𝑘−𝑛)𝑥 𝜏 ̃ 𝗑(𝑘)𝑒 𝖽𝑥 ∑ 𝜏∫ ] 0 [𝑘∈ℤ 𝜏

=

∑

̃ 𝗑(𝑘)

𝑘∈ℤ

2𝜋 1 𝑒𝑖 𝜏 (𝑘−𝑛)𝑥 𝖽𝑥 [𝜏 ∫ ] 0

𝜏

1 1 𝑖 2𝜋 (𝑘−𝑛)𝑥 𝜏 ̃ = 𝗑(𝑘) 𝑒 2𝜋 ∑ 𝜏 [ ] 𝑖 (𝑘 − 𝑛) 𝑘∈ℤ 𝜏

0

1 𝑖2𝜋(𝑘−𝑛) ̃ = 𝗑(𝑘) − 1] [𝑒 ∑ 𝑖2𝜋(𝑘 − 𝑛) 𝑘∈ℤ =

∑

𝑒𝑖2𝜋𝑥 − 1 𝑥→0 [ 𝑖2𝜋𝑥 ]

̄ − 𝑛) lim ̃ 𝛿(𝑘 𝗑(𝑘)

𝑘∈ℤ 𝗱 𝗱𝘅

̃ = 𝗑(𝑛) ̃ = 𝗑(𝑛)

𝑖2𝜋𝑥 − 1) (𝑒 𝗱 (𝑖2𝜋𝑥) 𝗱𝘅 𝑖2𝜋𝑥

𝑖2𝜋𝑒 𝑖2𝜋

|

by l'Hôpital's rule 𝑥=0

|𝑥=0

̃ = 𝗑(𝑛) ̃ = 𝐈𝗑(𝑛)

by definition of 𝐈 (Definition D.4 page 102)

✏ Theorem 5.2. t The Fourier Series adjoint operator 𝐅̂ ∗ is given by h 𝐅̂ ∗ = 𝐅̂ −1 m Tuesday 3rd October, 2017 3:01am UTC Copyright ©2017 Daniel J. Greenhoe


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page 68

Daniel J. Greenhoe

CHAPTER 5. FOURIER SERIES

✎PROOF: 𝜏

2𝜋

𝑛𝑥

̃ 𝖽𝑥 | 𝗒(𝑛) ⟩ℤ

by definition of 𝐅̂ Definition 5.1 page 65

2𝜋

𝑛𝑥

̃ ⟩ | 𝗒(𝑛)

by additivity property of ⟨△ | ▽⟩

̂ ̃ ⟩ℤ = 1 | 𝗒(𝑛) 𝗑(𝑥)𝑒−𝑖 𝜏 ⟨𝐅𝗑(𝑥) ⟨ √𝜏 ∫ 0 =

𝜏

1 √𝜏

∫ 0

𝗑(𝑥) ⟨𝑒−𝑖 𝜏

𝜏

=

∫ 0

𝗑(𝑥)

𝜏

= =

∫ 0

1 √𝜏

̃ |𝑒 ⟨𝗒(𝑛)

𝑛𝑥 −𝑖 2𝜋 𝜏

ℤ ∗

𝖽𝑥

⟩ℤ 𝖽𝑥

∗

̃ ] 𝖽𝑥 𝗑(𝑥) [𝐅̂ −1 𝗒(𝑛)

by property of ⟨△ | ▽⟩ by definition of 𝐅̂ −1 page 66

̂ −1 𝗒(𝑛) ̃ 𝗑(𝑥) | 𝐅 ⏟ ⟨ ⟩ 𝐅̂ ∗

ℝ

✏ The Fourier Series operator has several nice properties: 𝐅̂ is unitary 4 (Corollary 5.1 page 68). Because 𝐅̂ is unitary, it automatically has several other nice properties such as being isometric, and satisfying Parseval's equation, satisfying Plancheral's formula, and more (Corollary 5.2 page 68). Corollary 5.1. Let 𝐈 be the identity operator and let 𝐅̂ be the Fourier Series operator with adjoint 𝐅̂ ∗ . c o 𝐅̂ 𝐅̂ ∗ = 𝐅̂ ∗ 𝐅̂ = 𝐈 (𝐅̂ is unitary…and thus also normal and isometric) r This follows directly from the fact that 𝐅̂ ∗ = 𝐅̂ −1 (Theorem 5.2 (page 67)).

✎PROOF:

✏

Corollary 5.2. Let 𝐅̂ be the Fourier series operator, 𝐅̂ ∗ be its adjoint, and 𝐅̂ −1 be its inverse. 𝓡(𝐅)̂ = 𝓡(𝐅̂ −1 ) = 𝙇𝟤ℝ −1 = 1 ( UNITARY) ⦀𝐅̂ ⦀ = ⦀𝐅̂ ⦀ c −1 −1 ̂ ̂ ̂ ̂ o ( PARSEVAL'S EQUATION) ⟨𝐅𝒙 | 𝐅𝒚⟩ = ⟨𝐅 𝒙 | 𝐅 𝒚⟩ = ⟨𝒙 | 𝒚⟩ r −1 ̂ ̂ = ‖𝒙‖ ( PLANCHEREL'S FORMULA) ‖𝐅𝒙‖ = ‖𝐅 𝒙‖ −1 −1 ̂ ̂ ̂ ̂ ‖𝐅𝒙 − 𝐅𝒚‖ = ‖𝐅 𝒙 − 𝐅 𝒚‖ = ‖𝒙 − 𝒚‖ ( ISOMETRIC) These results follow directly from the fact that 𝐅̂ is unitary (Corollary 5.1 page 68) and from the properties of unitary operators (Theorem D.21 page 123). ✏

✎PROOF:

5.3

Fourier series for compactly supported functions

Theorem 5.3. The set t 1 𝑖 2𝜋 h 𝑒 𝜏 𝑛𝑥 𝑛∈ℤ | } { √𝜏 m is an ORTHONORMAL BASIS for all functions 𝖿(𝑥) with support in [0 ∶ 𝜏].

4

unitary operators: Definition D.15 page 122

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CHAPTER 6 DISCRETE TIME FOURIER TRANSFORM

6.1 Definition Definition 6.1. ̆ d The discrete-time Fourier transform 𝐅 of ⦅𝑥𝑛 ⦆𝑛∈ℤ is defined as −𝑖𝜔𝑛 𝟤 e ∀⦅𝑥𝑛 ⦆𝑛∈ℤ ∈𝓵ℝ [𝐅̆ ⦅𝑥𝑛 ⦆](𝜔) ≜ ∑ 𝑥𝑛 𝑒 f 𝑛∈ℤ

If we compare the definition of the Discrete Time Fourier Transform (Definition 6.1 page 69) to the definition of the Z-transform (Definition C.4 page 91), we see that the DTFT is just a special case of the more general Z-Transform, with 𝑧 = 𝑒𝑖𝜔 . If we imagine 𝑧 ∈ ℂ as a complex plane, then 𝑒𝑖𝜔 is a unit circle in this plane. The “frequency” 𝜔 in the DTFT is the unit circle in the much larger z-plane, as illustrated to the right.

ℑ[𝑧] 1

𝑧 = 𝑒𝑖𝜔 1

−1

ℜ[𝑧]

−1

6.2 Properties ̆ Proposition 6.1. Let 𝗑(𝜔) ≜ 𝐅̆ [⦅𝑥𝑛 ⦆](𝜔) be the DISCRETE-TIME FOURIER TRANSFORM (Definition 6.1 page 69) 𝟤 of a sequence ⦅𝑥𝑛 ⦆𝑛∈ℤ in 𝓵ℝ . p r 𝗑(𝜔) ̆ ̆ + 2𝜋𝑛) = 𝗑(𝜔 ∀𝑛 ∈ ℤ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ p PERIODIC with period 2𝜋

✎PROOF: ̆ + 2𝜋𝑛) = 𝗑(𝜔

∑

𝑥𝑚 𝑒−𝑖(𝜔+2𝜋𝑛)𝑚

𝑚∈ℤ

=

∑

=

∑

:1

𝑒−𝑖2𝜋𝑛𝑚 𝑥𝑚 𝑒−𝑖𝜔𝑚

𝑚∈ℤ

𝑥𝑚 𝑒

−𝑖𝜔𝑚

̆ = 𝗑(𝜔)

(6.1)

𝑚∈ℤ

✏

page 70

Daniel J. Greenhoe ℑ[𝑧]

CHAPTER 6. DISCRETE TIME FOURIER TRANSFORM

𝑧 = 𝑒𝑖𝜔

1

∑

ℎ𝑛

𝑛∈ℤ 1

−1

ℜ[𝑧]

𝜔

̂ ̃ 𝗁(−1) = 𝗁(𝜋) =0

−4𝜋

−1

−3𝜋

−2𝜋

−𝜋

0

𝜋

2𝜋

3𝜋

4𝜋

̂ be the Z-TRANSFORM (Definition C.4 page 91) and 𝗑(𝜔) ̆ Proposition 6.2. Let 𝗑(𝑧) the DISCRETE-TIME FOURIER TRANSFORM (Definition 6.1 page 69) of ⦅𝑥𝑛 ⦆. p 𝟤 , 𝑐∈ℝ r ̂ | ̆ | 𝑥 =𝑐 ⟺ = 𝑐} ⟺ 𝗑(𝜔) = 𝑐} ∀⦅𝑥𝑛 ⦆𝑛∈ℤ ∈𝓵ℝ {𝗑(𝑧) { ∑ 𝑛 𝑧=1 𝜔=0 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ p {𝑛∈ℤ } ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (2) z domain

(1) time domain

(3) frequency domain

✎PROOF: 1. Proof that (1) ⟹ (2): ̂ | 𝗑(𝑧) = 𝑥 𝑧−𝑛 ∑ 𝑛 | 𝑧=1 𝑛∈ℤ

=

∑

̂ (Definition C.4 page 91) by definition of 𝗑(𝑧) 𝑧=1

because 𝑧𝑛 = 1 for all 𝑛 ∈ ℤ

𝑥𝑛

𝑛∈ℤ

=𝑐

by hypothesis (1)

2. Proof that (2) ⟹ (3): ̆ | 𝗑(𝜔)

=

𝜔=0

∑

𝑥𝑛 𝑒−𝑖𝜔𝑛

𝑛∈ℤ

=

∑

𝑥𝑛 𝑧−𝑛

𝑛∈ℤ

̂ | = 𝗑(𝑧)

|

|

̆ by definition of 𝗑(𝜔) (Definition 6.1 page 69) 𝜔=0

𝑧=1

̂ (Definition C.4 page 91) by definition of 𝗑(𝑧)

𝑧=1

=𝑐

by hypothesis (2)

3. Proof that (3) ⟹ (1):

∑

𝑥𝑛 =

𝑛∈ℤ

∑


𝑛∈ℤ

|

𝜔=0

̆ = 𝗑(𝜔)

̆ by definition of 𝗑(𝜔) (Definition 6.1 page 69)

=𝑐

by hypothesis (3)

✏ Proposition 6.3. 1

1

📘 Chui (1992) page 123

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6.2. PROPERTIES

(−1)𝑛 𝑥𝑛 = 𝑐 ∑ 𝑛∈ℤ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Daniel J. Greenhoe

̂ 𝑧=−1 = 𝑐 ⟺ 𝗑(𝑧)| ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

page 71

̆ ⟺ 𝗑(𝜔)| 𝜔=𝜋 = 𝑐 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (3) in “frequency”

(2) in “𝑧 domain”

(1) in “time”

p r p

1 1 ℎ𝑛 + 𝑐 , ℎ −𝑐 ℎ2𝑛 , ℎ2𝑛+1 = ∑ ∑ ∑ 𝑛 ∑ (𝑛∈ℤ ) ( 2 (𝑛∈ℤ ) 2 (𝑛∈ℤ )) 𝑛∈ℤ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

⟺

(4) sum of even, sum of odd 𝟤 ∀𝑐∈ℝ, ⦅𝑥𝑛 ⦆𝑛∈ℤ , ⦅𝑦𝑛 ⦆𝑛∈ℤ ∈𝓵ℝ

✎PROOF: 1. Proof that (1) ⟹ (2): ̂ 𝑧=−1 = 𝗑(𝑧)|

∑


∑

(−1)𝑛 𝑥𝑛

𝑛∈ℤ

=

|

𝑧=−1

𝑛∈ℤ

=𝑐

by (1)

2. Proof that (2) ⟹ (3): ∑


𝑛∈ℤ

|

=

∑

(−1)𝑛 𝑥𝑛

∑

(−1)−𝑛 𝑥𝑛

𝑛∈ℤ

𝜔=𝜋

=

=

𝑛∈ℤ

∑

𝑧−𝑛 𝑥𝑛

𝑛∈ℤ

|

𝑧=−1

=𝑐

by (2)

3. Proof that (3) ⟹ (1): ∑

(−1)𝑛 𝑥𝑛 =

∑

(−1)−𝑛 𝑥𝑛

∑

𝑒−𝑖𝜔𝑛 𝑥𝑛

𝑛∈ℤ

𝑛∈ℤ

=

𝑛∈ℤ

|

𝜔=𝜋

=𝑐

by (3)

4. Proof that (2) ⟹ (4): (a) Define 𝐴 ≜

∑

ℎ2𝑛

𝐵≜

𝑛∈ℤ

∑

ℎ2𝑛+1 .

𝑛∈ℤ

(b) Proof that 𝐴 − 𝐵 = 𝑐: 𝑐=

∑

(−1)𝑛 𝑥𝑛

by (2)

𝑛∈ℤ

=

(−1)𝑛 𝑥𝑛 + (−1)𝑛 𝑥𝑛 ∑ ∑ 𝑛∈ℤe o ⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑛∈ℤ ⏟⏟⏟⏟⏟⏟⏟⏟⏟ even terms

=

∑

odd terms

2𝑛

(−1) 𝑥2𝑛 +

𝑛∈ℤ

=

𝑥 ∑ 2𝑛 𝑛∈ℤ ⏟ 𝐴

∑

(−1)2𝑛+1 𝑥2𝑛+1

𝑛∈ℤ

−

𝑥 ∑ 2𝑛+1 𝑛∈ℤ ⏟⏟⏟⏟⏟ 𝐵

≜𝐴−𝐵

by definitions of 𝐴 and 𝐵

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(c) Proof that 𝐴 + 𝐵 =

∑


𝑥𝑛 :

𝑛∈ℤ

∑

𝑥𝑛 =

𝑥𝑛 +

∑

𝑛 even

𝑛∈ℤ

=

∑

𝑥𝑛

𝑛 odd

𝑥 ∑ 2𝑛 𝑛∈ℤ ⏟

+

𝑥 ∑ 2𝑛+1 𝑛∈ℤ ⏟⏟⏟⏟⏟

𝐴

𝐵

=𝐴+𝐵

by definitions of 𝐴 and 𝐵

(d) This gives two simultaneous equations: 𝐴−𝐵 =𝑐 𝐴+𝐵 =

∑

𝑥𝑛

𝑛∈ℤ

(e) Solutions to these equations give

𝑥2𝑛 ≜ 𝐴

=

1 𝑥 +𝑐 ∑ 𝑛 2 (𝑛∈ℤ )

𝑥2𝑛+1 ≜ 𝐵

=

1 𝑥 −𝑐 ∑ 𝑛 2 (𝑛∈ℤ )

∑

𝑛∈ℤ

∑

𝑛∈ℤ

5. Proof that (2) ⟸ (4):

∑

(−1)𝑛 𝑥𝑛 =

𝑛∈ℤ

(−1)𝑛 𝑥𝑛 + (−1)𝑛 𝑥𝑛 ∑ ∑ 𝑛∈ℤe o ⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑛∈ℤ even terms

=

odd terms

2𝑛

∑

(−1) 𝑥2𝑛 +

∑

𝑥2𝑛 −

𝑛∈ℤ

=

(−1)2𝑛+1 𝑥2𝑛+1

𝑛∈ℤ

𝑛∈ℤ

=

∑

∑

𝑥2𝑛+1

𝑛∈ℤ

1 1 𝑥𝑛 + 𝑐 − 𝑥 −𝑐 ∑ ∑ 𝑛 2 (𝑛∈ℤ ) 2 (𝑛∈ℤ )

by (3)

=𝑐

✏ ̃ Lemma 6.1. Let 𝖿 (𝜔) be the DTFT (Definition 6.1 page 69) of a sequence ⦅𝑥𝑛 ⦆𝑛∈ℤ . l 2 2 𝟤 e ̆ ̆ ∀⦅𝑥𝑛 ⦆𝑛∈ℤ ∈𝓵ℝ = |𝗑(−𝜔)| ⟹ |𝗑(𝜔)| ⦅𝑥 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑛 ∈ ℝ⦆𝑛∈ℤ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ m EVEN

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✎PROOF: 2 ̆ ̂ 2 |𝑧=𝑒𝑖𝜔 |𝗑(𝜔)| = |𝗑(𝑧)|

̂ 𝗑̂∗ (𝑧)|𝑧=𝑒𝑖𝜔 = 𝗑(𝑧) =

=

=

∑ [𝑛∈ℤ ∑ [𝑛∈ℤ

𝑥𝑛 𝑧

−𝑛


∑∑

∗

∑ ] [𝑚∈ℤ ∑ ] [𝑚∈ℤ

𝑥𝑚 𝑧

∑[

𝑥𝑛 𝑥∗𝑚 𝑧−𝑛 (𝑧∗ )−𝑚

|𝑥𝑛 |2 +

∑[

|𝑥𝑛 |2 +

∑[

|𝑥𝑛 |2 +

∑[

|𝑥𝑛 |2 +

𝑛∈ℤ

=

∑[

|𝑥𝑛 |2 +

∑

𝑧=𝑒𝑖𝜔

𝑥𝑛 𝑥∗𝑚 𝑧−𝑛 (𝑧∗ )−𝑚 +

∑

𝑥𝑛 𝑥𝑚 𝑒𝑖𝜔(𝑚−𝑛) +

∑

𝑥𝑛 𝑥𝑚 𝑒𝑖𝜔(𝑚−𝑛) +

|𝑥𝑛 |2 + 2

∑

𝑥𝑛 𝑥∗𝑚 𝑧−𝑛 (𝑧∗ )−𝑚

𝑚𝑛 ∑

∑

𝑥𝑛 𝑥𝑚 2𝖼𝗈𝗌[𝜔(𝑚 − 𝑛)]

𝑚>𝑛

𝑛∈ℤ

=

𝑧=𝑒𝑖𝜔

∑

𝑚>𝑛

𝑛∈ℤ

=

]|

𝑚>𝑛

𝑛∈ℤ

=

|

𝑧=𝑒𝑖𝜔

𝑚>𝑛

𝑛∈ℤ

=

] |

𝑥∗𝑚 (𝑧∗ )−𝑚

𝑛∈ℤ 𝑚∈ℤ

=

−𝑛

∑∑

]

𝑥𝑛 𝑥𝑚 𝖼𝗈𝗌[𝜔(𝑚 − 𝑛)]

𝑛∈ℤ 𝑚>𝑛

𝑛∈ℤ

2 ̆ Since 𝖼𝗈𝗌 is real and even, then |𝗑(𝜔)| must also be real and even.

✏

2

̆ Theorem 6.1 (inverse DTFT). Let 𝗑(𝜔) be the DISCRETE-TIME FOURIER TRANSFORM (Definition 6.1 page 𝟤 −1 69) of a sequence ⦅𝑥𝑛 ⦆𝑛∈ℤ ∈ 𝓵 . Let 𝗑̃ be the inverse of 𝗑.̆ ℝ 𝛼+𝜋 t 1 𝑖𝜔𝑛 𝟤 h ̆ ̆ 𝗑(𝜔) ≜ 𝑥𝑛 𝑒−𝑖𝜔𝑛 ⟹ 𝑥𝑛 = 𝗑(𝜔)𝑒 𝖽𝜔 ∀𝛼∈ℝ ∀⦅𝑥𝑛 ⦆𝑛∈ℤ ∈𝓵ℝ ∑ { } ∫ 2𝜋 m { } 𝛼−𝜋 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑛∈ℤ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⦅𝑥𝑛 ⦆ = 𝐅̆ −1 𝐅̆ ⦅𝑥𝑛 ⦆

̆ 𝗑(𝜔) ≜ 𝐅̆ ⦅𝑥𝑛 ⦆

✎PROOF: 𝛼+𝜋

𝛼+𝜋

1 1 𝑖𝜔𝑛 ̆ 𝗑(𝜔)𝑒 𝖽𝜔 = 𝑥 𝑒−𝑖𝜔𝑚 𝑒𝑖𝜔𝑛 𝖽𝜔 ∑ 𝑚 ∫ 2𝜋 𝛼−𝜋 2𝜋 ∫ ] 𝛼−𝜋 [𝑚∈ℤ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

̆ by definition of 𝗑(𝜔)

̆ 𝗑(𝜔) 𝛼+𝜋

=

1 𝑥 𝑒−𝑖𝜔(𝑚−𝑛) 𝖽𝜔 ∑ 𝑚 2𝜋 ∫ 𝛼−𝜋 𝑚∈ℤ

=

1 𝑥 𝑒−𝑖𝜔(𝑚−𝑛) 𝖽𝜔 ∑ 𝑚∫ 2𝜋 𝑚∈ℤ 𝛼−𝜋

=

1 ̄ ] 𝑥 2𝜋 𝛿𝑚−𝑛 ∑ 𝑚[ 2𝜋 𝑚∈ℤ

𝛼+𝜋

= 𝑥𝑛 2

📘 J.S.Chitode (2009) page 3-95 ⟨(3.6.2)⟩



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✏ ̆ Theorem 6.2 (orthonormal quadrature conditions). 3 Let 𝗑(𝜔) be the DISCRETE-TIME FOURIER TRANS𝟤 ̄ FORM (Definition 6.1 page 69) of a sequence ⦅𝑥𝑛 ⦆𝑛∈ℤ ∈ 𝓵ℝ . Let 𝛿𝑛 be the KRONECKER DELTA FUNCTION at 𝑛 (Definition D.2 page 102). 𝟤 ̆ 𝗒∗̆ (𝜔) + 𝗑(𝜔 ̆ + 𝜋)𝗒∗̆ (𝜔 + 𝜋) = 0 𝑥 𝑦∗ = 0 ⟺ 𝗑(𝜔) ∀𝑛∈ℤ, ∀⦅𝑥𝑛 ⦆,⦅𝑦𝑛 ⦆∈𝓵ℝ ∑ 𝑚 𝑚−2𝑛 t 𝑚∈ℤ h 2 𝟤 ̆ ̆ + 𝜋)|2 = 2 + |𝗑(𝜔 ∀𝑛∈ℤ, ∀⦅𝑥𝑛 ⦆,⦅𝑦𝑛 ⦆∈𝓵ℝ |𝗑(𝜔)| m ∑ 𝑥𝑚 𝑥∗𝑚−2𝑛 = 𝛿𝑛̄ ⟺ 𝑚∈ℤ

Let 𝑧 ≜ 𝑒𝑖𝜔 .

✎PROOF:

1. Proof that 2

̆ 𝗒∗̆ (𝜔) + 𝗑(𝜔 ̆ + 𝜋)𝗒∗̆ (𝜔 + 𝜋): 𝑥𝑘 𝑦∗𝑘−2𝑛 𝑒−𝑖2𝜔𝑛 = 𝗑(𝜔) ] 𝑘∈ℤ

∑ [∑

𝑛∈ℤ

2

𝑥𝑘 𝑦∗𝑘−2𝑛 𝑒−𝑖2𝜔𝑛 ] 𝑘∈ℤ

∑ [∑

𝑛∈ℤ

=2

∑

𝑥𝑘

∑

𝑥𝑘

∑

𝑦∗𝑘−𝑛 𝑧−𝑛 +

∑

𝑦∗𝑚 𝑧−(𝑘−𝑚) +

∑

𝑥𝑘

∑

𝑥𝑘

∑

𝑥𝑘 𝑧−𝑘

𝑘∈ℤ

=

𝑦∗𝑘−𝑛 𝑧−𝑛 (1 + 𝑒𝑖𝜋𝑛 )

𝑥𝑘

𝑛∈ℤ

𝑛∈ℤ

𝑘∈ℤ

=

𝑦∗𝑘−𝑛 𝑧−𝑛

∑

∑

𝑘∈ℤ

=

∑

𝑛 even

𝑘∈ℤ

=

𝑦∗𝑘−2𝑛 𝑧−2𝑛

𝑛∈ℤ

𝑘∈ℤ

=2

∑

∑

∑

𝑥𝑘

𝑦∗𝑚 𝑧𝑚 +

𝑚∈ℤ

∑

∑

𝑦∗𝑚 𝑒−𝑖(𝜔+𝜋)(𝑘−𝑚)

where 𝑚 ≜ 𝑘 − 𝑛

𝑚∈ℤ

𝑘∈ℤ

∑

𝑦∗𝑘−𝑛 𝑧−𝑛 𝑒𝑖𝜋𝑛

∑

𝑛∈ℤ

𝑘∈ℤ

𝑚∈ℤ

𝑘∈ℤ

𝑥𝑘

𝑥𝑘 𝑒−𝑖(𝜔+𝜋)𝑘

∑

𝑦∗𝑚 𝑒+𝑖(𝜔+𝜋)𝑚

𝑚∈ℤ

𝑘∈ℤ ∗

=

∑

𝑥𝑘 𝑒

𝑘∈ℤ

−𝑖𝜔𝑘

∑ [𝑚∈ℤ

𝑦𝑚 𝑒

−𝑖𝜔𝑚

]

∗

+

∑

−𝑖(𝜔+𝜋)𝑘

𝑥𝑘 𝑒

𝑘∈ℤ

∑ [𝑚∈ℤ

𝑦𝑚 𝑒

−𝑖(𝜔+𝜋)𝑚

]

̆ 𝗒∗̆ (𝜔) + 𝗑(𝜔 ̆ + 𝜋)𝗒∗̆ (𝜔 + 𝜋) ≜ 𝗑(𝜔) ̆ 𝗒∗̆ (𝜔) + 𝗑(𝜔 ̆ + 𝜋)𝗒∗̆ (𝜔 + 𝜋) = 0: 2. Proof that ∑𝑚∈ℤ 𝑥𝑚 𝑦∗𝑚−2𝑛 = 0 ⟹ 𝗑(𝜔) 0=2


∑ [∑

𝑛∈ℤ

̆ 𝗒∗̆ (𝜔) + 𝗑(𝜔 ̆ + 𝜋)𝗒∗̆ (𝜔 + 𝜋) = 𝗑(𝜔)

by left hypothesis by item (1)

̆ 𝗒∗̆ (𝜔) + 𝗑(𝜔 ̆ + 𝜋)𝗒∗̆ (𝜔 + 𝜋) = 0: 3. Proof that ∑𝑚∈ℤ 𝑥𝑚 𝑦∗𝑚−2𝑛 = 0 ⟸ 𝗑(𝜔) 2

̆ 𝗒∗̆ (𝜔) + 𝗑(𝜔 ̆ + 𝜋)𝗒∗̆ (𝜔 + 𝜋) 𝑥𝑘 𝑦∗𝑘−2𝑛 𝑒−𝑖2𝜔𝑛 = 𝗑(𝜔) ] 𝑘∈ℤ

∑ [∑

𝑛∈ℤ

=0

by item (1) by right hypothesis

Thus by the above equation, ∑𝑛∈ℤ [∑𝑘∈ℤ 𝑥𝑘 𝑦∗𝑘−2𝑛 ] 𝑒−𝑖2𝜔𝑛 = 0. The only way for this to be true is if ∑𝑘∈ℤ 𝑥𝑘 𝑦∗𝑘−2𝑛 = 0. 3

📘 Daubechies (1992) pages 132–137 ⟨(5.1.20),(5.1.39)⟩

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Daniel J. Greenhoe

page 75

2 ̆ ̆ ′ + 𝜋)|2 = 2: 4. Proof that ∑𝑚∈ℤ 𝑥𝑚 𝑥∗𝑚−2𝑛 = 𝛿𝑛̄ ⟹ |𝗑(𝜔)| + |𝗑(𝜔 Let 𝑔𝑛 ≜ 𝑥𝑛 .

2=2

̄ 𝑒−𝑖2𝜔𝑛 𝛿𝑛∈ℤ

∑

𝑛∈ℤ

=2


by left hypothesis

∑ [∑

𝑛∈ℤ

̆ 𝗒∗̆ (𝜔) + 𝗑(𝜔 ̆ + 𝜋)𝗒∗̆ (𝜔 + 𝜋) = 𝗑(𝜔)

by item (1)

2 ̆ ̆ ′ + 𝜋)|2 = 2: + |𝗑(𝜔 5. Proof that ∑𝑚∈ℤ 𝑥𝑚 𝑥∗𝑚−2𝑛 = 𝛿𝑛̄ ⟸ |𝗑(𝜔)| Let 𝑔𝑛 ≜ 𝑥𝑛 .

2

̆ 𝗒∗̆ (𝜔) + 𝗑(𝜔 ̆ + 𝜋)𝗒∗̆ (𝜔 + 𝜋) 𝑥𝑘 𝑦∗𝑘−2𝑛 𝑒−𝑖2𝜔𝑛 = 𝗑(𝜔) ] 𝑘∈ℤ

by item (1)

∑ [∑

𝑛∈ℤ

=2

by right hypothesis

Thus by the above equation, ∑𝑛∈ℤ [∑𝑘∈ℤ 𝑥𝑘 𝑦∗𝑘−2𝑛 ] 𝑒−𝑖2𝜔𝑛 = 1. The only way for this to be true is if ∑𝑘∈ℤ 𝑥𝑘 𝑦∗𝑘−2𝑛 = 𝛿𝑛̄ .

✏

6.3 Derivatives Theorem 6.3. (A)

t h m

(C)

4

𝗱 [ 𝗱𝞈 ]

̆ Let 𝗑(𝜔) be the DTFT (Definition 6.1 page 69) of a sequence ⦅𝑥𝑛 ⦆𝑛∈ℤ .

𝑛

̆ | 𝗑(𝜔) = 0 𝜔=0

𝑛 𝗱 ̆ | [ 𝗱𝞈 ] 𝗑(𝜔) 𝜔=𝜋

⟺

= 0

⟺

∑

∑

𝑘𝑛 𝑥𝑘 = 0

𝑘∈ℤ 𝑘 𝑛

(−1) 𝑘 𝑥𝑘 = 0

(B)

∀𝑛∈𝕎

(D)

∀𝑛∈𝕎

𝑘∈ℤ

✎PROOF: 1. Proof that (𝐴) ⟹ (𝐵): 𝑛

𝗱 ̆ | 0 = [ 𝗱𝞈 ] 𝗑(𝜔) 𝜔=0 𝑛

𝗱 = [ 𝗱𝞈 ]

=

𝑥𝑘 𝑒−𝑖𝜔𝑘

𝑘∈ℤ

|

̆ by definition of 𝗑(𝜔) (Definition 6.1 page 69) 𝜔=0

𝑛

∑

𝑘∈ℤ

=

∑

by hypothesis (A)

∑

𝗱 −𝑖𝜔𝑘 𝑥𝑘 [ 𝗱𝞈 ] 𝑒

|

𝜔=0

𝑥𝑘 [(−𝑖)𝑛 𝑘𝑛 𝑒−𝑖𝜔𝑘 ]

𝑘∈ℤ

= (−𝑖)𝑛

∑

|

𝜔=0

𝑘𝑛 𝑥𝑘

𝑘∈ℤ 4

📘 Vidakovic (1999), pages 82–83, 📘 Mallat (1999), pages 241–242



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2. Proof that (𝐴) ⟸ (𝐵): 𝑛

𝑛

𝗱 𝗱 −𝑖𝜔𝑘 ̆ | = [ 𝗱𝞈 [ 𝗱𝞈 ] 𝗑(𝜔) ] ∑ 𝑥𝑘 𝑒 𝜔=0 | 𝑘∈ℤ

by definition of 𝗀̃ 𝜔=0

𝑛

𝗱 −𝑖𝜔𝑘 𝑥𝑘 [[ 𝗱𝞈 ] 𝑒 ]| 𝑘∈ℤ

=

∑

𝜔=0

𝑥𝑘 [(−𝑖)𝑛 𝑘𝑛 𝑒−𝑖𝜔𝑘 ] | 𝑘∈ℤ

=

∑

= (−𝑖)

𝑛

∑

𝜔=0

𝑛

𝑘 𝑥𝑘

𝑘∈ℤ

=0

by hypothesis (B)

3. Proof that (𝐶) ⟹ (𝐷): 𝑛

𝗱 ̆ | 0 = [ 𝗱𝞈 ] 𝗑(𝜔) 𝜔=𝜋 𝗱 = [ 𝗱𝞈 ]

=

=

∑

𝑥𝑘 𝑒−𝑖𝜔𝑘

𝑘∈ℤ

|

by definition of 𝗑̆ (Definition 6.1 page 69) 𝜔=𝜋

𝑛

∑

𝑘∈ℤ

=

𝑛

by hypothesis (C)


|

𝜔=𝜋

𝑥𝑘 [(−𝑖)𝑛 𝑘𝑛 𝑒−𝑖𝜔𝑘 ] | 𝑘∈ℤ ∑ ∑

𝑛 𝑛

𝜔=𝜋

𝑘

𝑥𝑘 [(−𝑖) 𝑘 (−1) ]

𝑘∈ℤ

= (−𝑖)𝑛

∑

(−1)𝑘 𝑘𝑛 𝑥𝑘

𝑘∈ℤ

4. Proof that (𝐶) ⟸ (𝐷): 𝑛

𝑛

𝗱 𝗱 −𝑖𝜔𝑘 ̆ | = [ 𝗱𝞈 [ 𝗱𝞈 ] 𝗑(𝜔) ] ∑ 𝑥𝑘 𝑒 𝜔=𝜋 | 𝑘∈ℤ

=

=

𝜔=𝜋

𝑛

∑

𝑘∈ℤ

=

by definition of 𝗑̆ (Definition 6.1 page 69)


|

𝜔=𝜋

𝑥𝑘 [(−𝑖)𝑛 𝑘𝑛 𝑒−𝑖𝜔𝑘 ] | 𝑘∈ℤ ∑

∑

𝑛 𝑛

𝜔=𝜋

𝑘

𝑥𝑘 [(−𝑖) 𝑘 (−1) ]

𝑘∈ℤ

= (−𝑖)𝑛

∑

(−1)𝑘 𝑘𝑛 𝑥𝑘

𝑘∈ℤ

=0

by hypothesis (D)

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6.4. FREQUENCY RESPONSE

Daniel J. Greenhoe

page 77

6.4 Frequency Response The pole zero locations of a digital filter determine the magnitude and phase frequency response of the digital filter.5 This can be seen by representing the poles and zeros vectors in the complex z-plane. Each of these vectors has a magnitude 𝑀 and a direction 𝜃. Also, each factor (𝑧 − 𝑧𝑖 ) and (𝑧 − 𝑝𝑖 ) can be represented as vectors as well (the difference of two vectors). Each of these factors can be represented by a magnitude/phase factor 𝑀𝑖 𝑒𝑖𝜃𝑖 . The overall magnitude and phase of 𝐻(𝑧) can then be analyzed. Take the following filter for example:

𝑏0 + 𝑏1 𝑧−1 + 𝑏2 𝑧−2

𝐻(𝑧) =

1 + 𝑎1 𝑧−1 + 𝑎2 𝑧−2 (𝑧 − 𝑧1 )(𝑧 − 𝑧2 ) = (𝑧 − 𝑝1 )(𝑧 − 𝑝2 ) 𝑀1 𝑒𝑖𝜃1 𝑀2 𝑒𝑖𝜃2 = 𝑀3 𝑒𝑖𝜃3 𝑀4 𝑒𝑖𝜃4 𝑀1 𝑀2 𝑒𝑖𝜃1 𝑒𝑖𝜃2 = ( 𝑀3 𝑀4 ) ( 𝑒𝑖𝜃3 𝑒𝑖𝜃4 )

This is illustrated in Figure 6.1 (page 77). The unit circle represents frequency in the Fourier domain. The frequency response of a filter is just a rotating vector on this circle. The magnitude response of the filter is just then a vector sum. For example, the magnitude of any 𝐻(𝑧) is as follows:

|𝐻(𝑧)| =

|(𝑧 − 𝑧1 )| |(𝑧 − 𝑧2 )| |(𝑧 − 𝑝1 )| |(𝑧 − 𝑝2 )| ℑ[𝑧] 1

𝑧=𝑒

𝑖𝜔

×

𝑧 − 𝑝1

𝑝1 −1

𝜔

𝑧1 𝑧 − 𝑧1

𝑧

1

𝑝2 ×

ℜ[𝑧]

𝑧 − 𝑧2 𝑧 − 𝑝2 𝑧2 −1

Figure 6.1: Vector response of digital filter

5

📘 Cadzow (1987), pages 90–91



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Part I Appendices

APPENDIX A NORMED ALGEBRAS

A.1 Algebras All linear spaces are equipped with an operation by which vectors in the spaces can be added together. Linear spaces also have an operation that allows a scalar and a vector to be “multiplied” together. But linear spaces in general have no operation that allows two vectors to be multiplied together. A linear space together with such an operator is an algebra.1 There are many many possible algebras—many more than one can shake a stick at, as indicated by Michiel Hazewinkel in his book, Handbook of Algebras: “Algebra, as we know it today (2005), consists of many different ideas, concepts and results. A reasonable estimate of the number of these different items would be somewhere between 50,000 and 200,000. Many of these have been named and many more could (and perhaps should) have a “name” or other convenient designation.”2

Definition A.1. 3 Let 𝑨 be an ALGEBRA. d e An algebra 𝑨 is unital if ∃𝑢 ∈ 𝑨 such that f

𝑢𝑥 = 𝑥𝑢 = 𝑥

∀𝑥 ∈ 𝑨

Definition A.2. 4 Let 𝑨 be an UNITAL ALGEBRA (Definition A.1 page 81) with unit 𝑒. 𝜎(𝑥) ≜ {𝜆 ∈ ℂ|𝜆𝑒 − 𝑥 is not invertible} . d The spectrum of 𝑥 ∈ 𝑨 is e The resolvent of 𝑥 ∈ 𝑨 is 𝜌𝑥 (𝜆) ≜ (𝜆𝑒 − 𝑥)−1 f The spectral radius of 𝑥 ∈ 𝑨 is 𝑟(𝑥) ≜ sup {|𝜆||𝜆 ∈ 𝜎(𝑥)} .

1

📘 Fuchs (1995), page 2 📘 Hazewinkel (2000), page v 3 📘 Folland (1995) page 1 4 📘 Folland (1995) pages 3–4 2

∀𝜆∉𝜎(𝑥).

page 82

A.2

Daniel J. Greenhoe

APPENDIX A. NORMED ALGEBRAS

Star-Algebras

Definition A.3. 5 Let 𝑨 be an ALGEBRA. The pair (𝑨, ∗) is a ∗-algebra if ∗ 1. (𝑥 + 𝑦) = 𝑥∗ + 𝑦∗ ∀𝑥,𝑦∈𝑨 ( DISTRIBUTIVE) ∗ ∗ d 2. (𝛼𝑥) = 𝛼𝑥 ̄ ∀𝑥∈𝑨, 𝛼∈ℂ ( CONJUGATE LINEAR) e ∗ ∗ ∗ 3. (𝑥𝑦) = 𝑦 𝑥 ∀𝑥,𝑦∈𝑨 ( ANTIAUTOMORPHIC) f ∗∗ 4. 𝑥 = 𝑥 ∀𝑥∈𝑨 ( INVOLUTORY) The operator ∗ is called an involution on the algebra 𝑨. Proposition A.1. 6 Let (𝑨, ∗) be an UNITAL ∗-ALGEBRA. ∗ p 1. 𝑥 is INVERTIBLE r 𝑥 is invertible ⟹ { 2. (𝑥∗ )−1 = (𝑥−1 )∗ p ✎PROOF:

∀𝑥∈𝑨

and and and

and

∀𝑥∈𝑨

Let 𝑒 be the unit element of (𝑨, ∗).

1. Proof that 𝑒∗ = 𝑒: ∗∗

𝑥 𝑒 ∗ = (𝑥 𝑒 ∗ )

by involutory property of Definition A.3 page 82

∗ ∗∗ ∗

= (𝑥 𝑒 ) ∗ = (𝑥∗ 𝑒)

by antiautomorphic property of Definition A.3 page 82 by involutory property of Definition A.3 page 82

∗ ∗

= (𝑥 ) =𝑥

by definition of 𝑒 by involutory property of Definition A.3 page 82 ∗∗

𝑒∗ 𝑥 = (𝑒∗ 𝑥)

by involutory property of Definition A.3 page 82

∗ ∗

= (𝑒∗∗ 𝑥 ) ∗ = (𝑒 𝑥 ∗ )

by antiautomorphic property of Definition A.3 page 82 by involutory property of Definition A.3 page 82

∗ ∗

= (𝑥 ) =𝑥 −1

2. Proof that (𝑥∗ )

by definition of 𝑒 by involutory property of Definition A.3 page 82 ∗

= (𝑥−1 ) : ∗

−1 ∗ −1 (𝑥 ) (𝑥 ) = [𝑥 (𝑥 )] = 𝑒∗

∗

by item (1) page 82

=𝑒 −1 ∗

∗

∗ −1 (𝑥 ) (𝑥 ) = [𝑥 𝑥] = 𝑒∗

by item (1) page 82

=𝑒

✏ Definition A.4. 7 Let (𝑨, ‖⋅‖) be a ∗-ALGEBRA (Definition A.3 page 82). An element 𝑥 ∈ 𝑨 is hermitian or self-adjoint if 𝑥∗ = 𝑥. d e An element 𝑥 ∈ 𝑨 is normal if 𝑥𝑥∗ = 𝑥∗ 𝑥. f An element 𝑥 ∈ 𝑨 is a projection if 𝑥𝑥 = 𝑥 ( INVOLUTORY) and 𝑥∗ = 𝑥 ( HERMITIAN). 5

📘 Rickart (1960), page 178, 📓 Gelfand and Naimark (1964), page 241 📘 Folland (1995) page 5 7 📘 Rickart (1960), page 178, 📓 Gelfand and Naimark (1964), page 242 6

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A.2. STAR-ALGEBRAS

Daniel J. Greenhoe

page 83

8

Let (𝑨, ‖⋅‖) be a ∗-ALGEBRA (Definition A.3 page 82). 𝑥 + 𝑦 = (𝑥 + 𝑦)∗ (𝑥 + 𝑦 is self adjoint) ⎧ ⎪ ∗ ∗ ∗ = (𝑥 ) (𝑥∗ is self adjoint) t ⎪ 𝑥 ∗ ∗ ∗ h ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑥 = 𝑥 and 𝑦 = 𝑦 ⟹ ⎨ ⏟⏟⏟⏟⏟⏟⏟ 𝑥𝑦 = (𝑥𝑦) ⟺ 𝑥𝑦 = 𝑦𝑥 ⏟⏟⏟ m ⎪ 𝑥 and 𝑦 are hermitian ⎪ commutative ⎩ (𝑥𝑦) is hermitian

Theorem A.1.

✎PROOF: (𝑥 + 𝑦)∗ = 𝑥∗ + 𝑦∗

by Definition A.3 page 82

=𝑥+𝑦

by left hypothesis

∗

∗ (𝑥 ) = 𝑥


Proof that 𝑥𝑦 = (𝑥𝑦)∗ ⟹ 𝑥𝑦 = 𝑦𝑥 𝑥𝑦 = (𝑥𝑦)∗

by left hypothesis

∗ ∗

=𝑦 𝑥


= 𝑦𝑥

by left hypothesis

Proof that 𝑥𝑦 = (𝑥𝑦)∗ ⟸ 𝑥𝑦 = 𝑦𝑥 (𝑥𝑦)∗ = (𝑦𝑥)∗

by left hypothesis

∗ ∗

=𝑥 𝑦


= 𝑥𝑦

by left hypothesis

✏ 9

Let (𝑨, ‖⋅‖) be a ∗-ALGEBRA (Definition A.3 page 82). 1 The real part of 𝑥 is defined as ℜ𝑥 ≜ 𝑥 + 𝑥∗ ) 2( 1 𝑥 − 𝑥∗ ) The imaginary part of 𝑥 is defined as ℑ𝑥 ≜ ( 2𝑖

Definition A.5 (Hermitian components). d e f

Theorem A.2. 10 Let (𝑨, ∗) be a ∗-ALGEBRA (Definition A.3 page 82). t ℜ𝑥 = (ℜ𝑥)∗ ∀𝑥∈𝑨 (ℜ𝑥 is hermitian) h ∗ ∀𝑥∈𝑨 (ℑ𝑥 is hermitian) m ℑ𝑥 = (ℑ𝑥) ✎PROOF: ∗ 1 (ℜ𝑥)∗ = ( (𝑥 + 𝑥∗ )) 2 1 ∗ = (𝑥 + 𝑥∗∗ ) 2 1 = (𝑥∗ + 𝑥) 2 = ℜ𝑥 ∗ 1 (ℑ𝑥)∗ = ( (𝑥 − 𝑥∗ )) 2𝑖

by Definition A.5 page 83 by Definition A.3 page 82 by Definition A.3 page 82 by Definition A.5 page 83 by Definition A.5 page 83

8

📘 Michel and Herget (1993) page 429 📘 Michel and Herget (1993) page 430, 📘 Rickart (1960), page 179, 📓 Gelfand and Naimark (1964), page 242 10 📘 Michel and Herget (1993) page 430, 📘 Halmos (1998) page 42 9



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Daniel J. Greenhoe


1 ∗ 𝑥 − 𝑥∗∗ ) 2𝑖 ( 1 = (𝑥∗ − 𝑥) 2𝑖 = ℑ𝑥 =

by Definition A.3 page 82 by Definition A.3 page 82 by Definition A.5 page 83

✏ Theorem A.3 (Hermitian representation). 11 Let (𝑨, ∗) be a ∗-ALGEBRA (Definition A.3 page 82). t h 𝑎 = 𝑥 + 𝑖𝑦 ⟺ 𝑥 = ℜ𝑎 and 𝑦 = ℑ𝑎 m

✎PROOF: Proof that 𝑎 = 𝑥 + 𝑖𝑦 ⟹ 𝑥 = ℜ𝑎

and 𝑦 = ℑ𝑎:

𝑎 = 𝑥 + 𝑖𝑦

by left hypothesis ∗

∗

𝑎 = (𝑥 + 𝑖𝑦)

⟹

∗

= 𝑥 − 𝑖𝑦

∗

= 𝑥 − 𝑖𝑦 ⟹

𝑥 = 𝑎 − 𝑖𝑦 𝑥 = 𝑎 + 𝑖𝑦

⟹

𝑥 + 𝑥 = 𝑎 + 𝑎∗

⟹

∗

by solving for 𝑥 in 𝑎∗ = 𝑥 − 𝑖𝑦 equation by adding previous 2 equations

𝑖𝑦 = 𝑎 − 𝑥

by solving for 𝑖𝑦 in 𝑎 = 𝑥 + 𝑖𝑦 equation

∗

𝑖𝑦 + 𝑖𝑦 = 𝑎 − 𝑎 1 𝑦 = (𝑎 − 𝑎 ∗ ) 2𝑖 = ℑ𝑎

Proof that 𝑎 = 𝑥 + 𝑖𝑦 ⟸ 𝑥 = ℜ𝑎

by definition of ℜ (Definition A.5 page 83)

by solving for 𝑖𝑦 in 𝑎 = 𝑥 + 𝑖𝑦 equation by adding previous 2 equations by solving for 𝑖𝑦 in previous equations by definition of ℑ (Definition A.5 page 83)

and 𝑦 = ℑ𝑎:

𝑥 + 𝑖𝑦 = ℜ𝑎 + 𝑖 ℑ𝑎 1 1 = (𝑎 + 𝑎∗ ) + 𝑖 (𝑎 − 𝑎∗ ) 2 2𝑖 ⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟ ℜ𝑎

by Theorem A.2 page 83

by solving for 𝑥 in previous equation

𝑖𝑦 = −𝑎 + 𝑥

⟹


2𝑥 = 𝑎 + 𝑎 1 𝑥 = (𝑎 + 𝑎∗ ) 2 = ℜ𝑎

∗

⟹

(Definition A.4 page 82)

by solving for 𝑥 in 𝑎 = 𝑥 + 𝑖𝑦 equation

∗

⟹

by definition of adjoint

by right hypothesis by definition of ℜ and ℑ (Definition A.5 page 83)

ℑ𝑎

:0 1 1 ∗ 1 1 ∗ = ( 𝑎 + 𝑎) + 𝑎− 𝑎 ) ( 2 2 2 2 =𝑎

✏ 11

📘 Michel and Herget (1993) page 430, 📘 Rickart (1960), page 179, 📓 Gelfand and Neumark (1943b), page 7

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A.3. NORMED ALGEBRAS

Daniel J. Greenhoe

page 85

A.3 Normed Algebras Definition A.6. 12 Let 𝑨 be an algebra. d The pair (𝑨, ‖⋅‖) is a normed algebra if e ∀𝑥, 𝑦 ∈ 𝑨 (multiplicative condition) ‖𝑥𝑦‖ ≤ ‖𝑥‖ ‖𝑦‖ f A normed algebra (𝑨, ‖⋅‖) is a Banach algebra if (𝑨, ‖⋅‖) is also a Banach space. Proposition A.2. p r (𝑨, ‖⋅‖) is a normed algebra p

multiplication is continuous in (𝑨, ‖⋅‖)

⟹

✎PROOF: 1. Define 𝖿(𝑥) ≜ 𝑧𝑥. That is, the function 𝖿 represents multiplication of 𝑥 times some arbitrary value 𝑧. 2. Let 𝛿 ≜ ‖𝑥 − 𝑦‖ and 𝜖 ≜ ‖𝖿 (𝑥) − 𝖿 (𝑦)‖. 3. To prove that multiplication (𝖿) is continuous with respect to the metric generated by ‖⋅‖, we have to show that we can always make 𝜖 arbitrarily small for some 𝛿 > 0. 4. And here is the proof that multiplication is indeed continuous in (𝑨, ‖⋅‖): ‖𝖿(𝑥) − 𝖿 (𝑦)‖ = ‖𝑧𝑥 − 𝑧𝑦‖

by definition of 𝖿

= ‖𝑧(𝑥 − 𝑦)‖ ≤ ‖𝑧‖ ‖𝑥 − 𝑦‖


= ‖𝑧‖ 𝛿

by definition of 𝛿

≤𝜖

for some value of 𝛿 > 0

✏ Theorem A.4 (Gelfand-Mazur Theorem). t (𝑨, ‖⋅‖) is a Banach algebra h m every nonzero 𝑥 ∈ 𝑨 is invertible }

13

Let ℂ be the field of complex numbers. ⟹

𝑨≡ℂ

(𝑨 is isomorphic to ℂ)

A.4 C* Algebras Definition A.7. 14 The triple (𝑨, ‖⋅‖ , ∗) is a 𝐶 ∗ algebra if d 1. (𝑨, ‖⋅‖) is a Banach algebra e (𝑨, 2. ∗) is a ∗-algebra f 2 ∗ 3. ‖𝑥 𝑥‖ = ‖𝑥‖ ∀𝑥 ∈ 𝑨. Theorem A.5. 15 Let 𝑨 be an algebra. t h (𝑨, ‖⋅‖ , ∗) is a 𝐶 ∗ algebra ⟹ m

and and

∗

‖𝑥 ‖ = ‖𝑥‖

12

📘 Rickart (1960), page 2, 📘 Berberian (1961) page 103 ⟨Theorem IV.9.2⟩ 📘 Folland (1995) page 4, 📃 Mazur (1938) ⟨(statement)⟩, 📃 Gelfand (1941) ⟨(proof)⟩ 14 📘 Folland (1995) page 1, 📓 Gelfand and Naimark (1964), page 241, 📃 Gelfand and Neumark (1943a), 📓 Gelfand and Neumark (1943b) 15 📘 Folland (1995) page 1, 📓 Gelfand and Neumark (1943b), page 4, 📃 Gelfand and Neumark (1943a) 13



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Daniel J. Greenhoe


✎PROOF: 1 ‖𝑥‖2 ‖𝑥‖ 1 = 𝑥∗ 𝑥‖ ‖𝑥‖ ‖ 1 ≤ 𝑥∗ ‖𝑥‖ ‖𝑥‖ ‖ ‖ = ‖𝑥∗ ‖

‖𝑥‖ =

∗ ∗∗ ‖𝑥 ‖ ≤ ‖𝑥 ‖ = ‖𝑥‖

by definition of 𝐶 ∗ -algebra Definition A.7 page 85 by definition of normed algebra Definition A.6 page 85

by previous result by involution property Definition A.3 page 82

✏

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APPENDIX B CALCULUS

Definition B.1. Let ℝ be the set of real numbers, ℬ the set of BOREL SETS on ℝ, and 𝜇 the standard BOREL MEASURE on ℬ. Let ℝℝ be as in Definition 4.1 page 49. The space of Lebesgue square-integrable functions 𝙇𝟤(ℝ,ℬ,𝜇) (or 𝙇𝟤ℝ ) is defined as 1

d e f

⎫ ⎧ 2 ⎪ ⎪ 𝖽𝜇 < ∞⎬. 𝙇𝟤ℝ ≜ 𝙇𝟤(ℝ,ℬ,𝜇) ≜ ⎨𝖿 ∈ ℝℝ | |𝖿|2 (∫ ) ℝ ⎪ ⎪ ⎩ ⎭ 𝟤 The standard inner product ⟨△ | ▽⟩ on 𝙇ℝ is defined as ⟨𝖿(𝑥) | 𝗀(𝑥)⟩ ≜

∫ ℝ

𝖿(𝑥)𝗀∗ (𝑥) 𝖽𝑥. 1

The standard norm ‖⋅‖ on 𝙇𝟤ℝ is defined as ‖𝖿(𝑥)‖ ≜ ⟨𝖿(𝑥) | 𝖿(𝑥)⟩ 2 Definition B.2. Let 𝖿(𝑥) be a FUNCTION in ℝℝ . d 𝖽 𝖿(𝑥 + 𝜀) − 𝖿(𝑥) e 𝖿(𝑥) ≜ 𝖿 ′ (𝑥) ≜ lim 𝜀→0 𝜀 f 𝖽𝑥 Proposition B.1. (1). 𝖿 (𝑥) is CONTINUOUS p ⎧ ⎪ (2). 𝖿⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (𝑎 + 𝑥) = 𝖿(𝑎 − 𝑥) r ⎨ p ⎪ SYMMETRIC about a point 𝑎 ⎩

and

⎫ ⎪ ⎬ ⟹ { ⎪ ⎭

(1). (2).

𝖿 ′ (𝑎 + 𝑥) = −𝖿 ′ (𝑎 − 𝑥) 𝖿 ′ (𝑎) = 0

( ANTI-SYMMETRIC about 𝑎)

✎PROOF: 1 𝖿 ′ (𝑎 + 𝑥) = lim [𝖿(𝑎 + 𝑥 + 𝜀) − 𝖿 (𝑎 + 𝑥 − 𝜀)] 𝜀→0 𝜀 1 = lim [𝖿(𝑎 − 𝑥 − 𝜀) − 𝖿 (𝑎 − 𝑥 + 𝜀)] 𝜀→0 𝜀 1 = − lim [𝖿(𝑎 − 𝑥 + 𝜀) − 𝖿(𝑎 − 𝑥 − 𝜀)] 𝜀→0 𝜀 = −𝖿(𝑎 − 𝑥) 1 1 𝖿 ′ (𝑎) = 𝖿 ′ (𝑎 + 0) + 𝖿 ′ (𝑎 − 0) 2 2 1 = [𝖿 ′ (𝑎 + 0) − 𝖿 ′ (𝑎 + 0)] 2

by hpothesis (2)

by previous result

}

page 88

Daniel J. Greenhoe

APPENDIX B. CALCULUS

=0

✏ Lemma B.1. ⎧ l ⎪ 𝖽 e 𝖿(𝑥) is INVERTIBLE ⟹ ⎨ 𝖿 −1 (𝑦) = 𝖽𝑦 m ⎪ ⎩

⎫ ⎪ ⎬ 𝖽 𝖿 𝖿 −1 (𝑦)] ⎪ 𝖽𝑥 [ ⎭ 1

✎PROOF: 𝖿 −1 (𝑦 + 𝜀) − 𝖿 −1 (𝑦) 𝖽 −1 𝖿 (𝑦) ≜ lim 𝜀→0 𝖽𝑦 𝜀 | | 1 | = lim 𝛿→0 𝖿(𝑥 + 𝛿) − 𝖿(𝑥) | | ] |𝑥≜𝖿 −1 (𝑦) [ 𝛿 1

≜

𝖽 𝖿(𝑥) | −1 𝖽𝑥 𝑥≜𝖿 (𝑦)

1

=

by definition of

𝖽 𝖽𝑦

(Definition B.2 page 87)

𝛥𝑦 𝛥𝑥 = because in the limit, 𝛥𝑥 ( 𝛥𝑦 )

by definition of

𝖽 𝖽𝑥

−1

(Definition B.2 page 87)

because 𝑥 ≜ 𝖿 −1 (𝑦)

𝖽 𝖿 𝖿 −1 (𝑦)] 𝖽𝑥 [

✏ Theorem B.1.

1

Let 𝖿 be a continuous function in 𝙇𝟤ℝ and 𝖿 (𝑛) the 𝑛th derivative of 𝖿 .

𝑛 𝑛 t 𝑛 (𝑛) h 𝖿 𝑥𝑘 𝖽𝑥1 𝖽𝑥2 ⋯ 𝖽𝑥𝑛 = (−1)𝑛−𝑘 ( )𝖿(𝑘) ∑ ∑ ∫ 𝑛 𝑘 m [0∶1) (𝑘=1 ) 𝑘=0

✎PROOF:

∀𝑛 ∈ ℕ

Proof by induction:

1. Base case …proof for 𝑛 = 1 case:

∫ [0∶1)

𝖿 (1) (𝑥) 𝖽𝑥 = 𝖿 (1) − 𝖿 (0)

by Fundamental theorem of calculus

1 1 = (−1)1+1 ( )𝖿(1) + (−1)1+0 ( )𝖿(0) 0 1 1

=

𝑛 (−1)𝑛−𝑘 ( )𝖿 (𝑘) ∑ 𝑘 𝑘=0

1

📘 Chui (1992) page 86 ⟨item (ii)⟩, 📘 Prasad and Iyengar (1997) pages 145–146 ⟨Theorem 6.2 (b)⟩

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2. Induction step …proof that 𝑛 case ⟹ 𝑛 + 1 case: ⎛ 𝑛+1 ⎞ 𝖿 (𝑛+1) ⎜ 𝑥𝑘 ⎟ 𝖽𝑥1 𝖽𝑥2 ⋯ 𝖽𝑥𝑛+1 ∫ ⎜∑ ⎟ [0∶1)𝑛+1 ⎝𝑘=1 ⎠ 𝑛

1

=

𝖿 (𝑛+1) 𝑥𝑛+1 + 𝑥 𝖽𝑥 𝖽𝑥 𝖽𝑥 ⋯ 𝖽𝑥𝑛 ∑ 𝑘 ) 𝑛+1 ] 1 2 ∫ 𝑛 [∫ ( 0 [0∶1) 𝑘=1 𝑥

=1

𝑛+1 𝑛 ⎡ (𝑛) ⎤ ⎢𝖿 ⎥ 𝖽𝑥1 𝖽𝑥2 ⋯ 𝖽𝑥𝑛 = 𝑥𝑛+1 + 𝑥𝑘 ∑ )| ∫ ⎥ ( [0∶1)𝑛 ⎢ 𝑘=1 𝑥𝑛+1 =0 ⎦ ⎣

𝑛

=

∑

𝑛

𝑛−𝑘

(−1)

𝑘=0

𝑛 𝑛−𝑘 𝑛 (𝑘)𝖿(𝑘 + 1) − ∑(−1) (𝑘)𝖿(𝑘) 𝑘=0

𝑚=𝑛+1

=

𝑛

𝖿 (𝑛) 1 + 𝑥𝑘 − 𝖿 (𝑛) 0 + 𝑥 𝖽𝑥 𝖽𝑥 ⋯ 𝖽𝑥𝑛 ∑ ∑ 𝑘 )] 1 2 ∫ 𝑛 [ ( ) ( [0∶1) 𝑘=1 𝑘=1 𝑛

=

by Fundamental theorem of calculus

∑

by induction hypothesis

𝑛

(−1)

𝑛−𝑚+1

𝑚=1

𝑛 𝑛−𝑘 𝑛 (𝑚 − 1)𝖿 (𝑚) + ∑(−1)(−1) (𝑘)𝖿 (𝑘) 𝑘=0

𝑛

where 𝑚 ≜ 𝑘 + 1 ⟹ 𝑘 = 𝑚 − 1 𝑛

𝑛 𝑛+1 𝑛−𝑘+1 𝑛 = 𝖿(𝑛 + 1) + (−1)𝑛−𝑘+1 ( )𝖿(𝑘)] + [(−1) 𝖿(0) + ∑(−1) (𝑘)𝖿(𝑘)] ∑ 𝑘 − 1 [ 𝑘=1 𝑘=1 𝑛

= 𝖿 (𝑛 + 1) + (−1)

𝑛+1

𝖿 (0) +

𝑛 𝑛 (−1)𝑛−𝑘+1 [( + ( )]𝖿(𝑘) ) ∑ 𝑘−1 𝑘 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑘=1

use Stifel formula 𝑛

𝑛+1 𝑛+1 𝑛+1 = (−1)0 ( 𝖿(𝑛 + 1) + (−1)𝑛+1 ( 𝖿(0) + (−1)𝑛−𝑘+1 ( 𝖿(𝑘) ) ) ∑ 𝑛+1 0 𝑘 ) 𝑘=1

by Stifel formula

𝑛+1

=

𝑛+1 (−1)𝑛−𝑘+1 ( 𝖿 (𝑘) 𝑘 ) 𝑘=0

∑

✏ Some proofs invoke differentiation multiple times. This is simplified thanks to the Leibniz rule, also called the generalized product rule (GPR, next lemma). The Leibniz rule is remarkably similar in form to the binomial theorem. Lemma B.2 (Leibniz rule). for 𝑛 = 0, 1, 2, …, and (𝑘𝑛) ≜

2

Let 𝖿(𝑥), 𝗀(𝑥) ∈ 𝙇𝟤ℝ with derivatives 𝖿 (𝑛) (𝑥) ≜

𝑛! (𝑛−𝑘)!𝑘!

d𝑛 𝖿(𝑥) and 𝗀(𝑛) (𝑥) d𝑥𝑛

≜

d𝑛 𝗀(𝑥) d𝑥𝑛

(binomial coefficient). Then

𝑛 l d𝑛 𝑛 (𝑘) (𝑛−𝑘) e [𝖿(𝑥)𝗀(𝑥)] = (𝑥) ( )𝖿 (𝑥)𝗀 𝑛 ∑ 𝑘 m d𝑥 𝑘=0

2

📘 Ben-Israel and Gilbert (2002) page 154, 📘 Leibniz (1710)



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APPENDIX B. CALCULUS

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APPENDIX C OPERATIONS ON SEQUENCES

C.1 Z-transform Definition C.1. 1 Let 𝑋 𝑌 be the set of all functions from a set 𝑌 to a set 𝑋. Let ℤ be the set of integers. d A function 𝖿 in 𝑋 𝑌 is a sequence over 𝑋 if 𝑌 = ℤ. e f A sequence may be denoted in the form ⦅𝑥𝑛 ⦆𝑛∈ℤ or simply as ⦅𝑥𝑛 ⦆. Definition C.2. 2 Let (𝔽 , +, ⋅, 0, 1) be a field. The space of all absolutely square summable sequences 𝓵𝔽𝟤 over 𝔽 is defined as d e 2 𝓵𝔽𝟤 ≜ ⦅𝑥𝑛 ⦆𝑛∈ℤ | 𝑥𝑛 | < ∞ f | ∑ } { 𝑛∈ℤ 𝟤 𝟤 The space 𝓵ℝ is an example of a separable Hilbert space. In fact, 𝓵ℝ is the only separable Hilbert 𝟤 space in the sense that all separable Hilbert spaces are isomorphically equivalent. For example, 𝓵ℝ is isomorphic to 𝙇𝟤ℝ , the space of all absolutely square Lebesgue integrable functions.

Definition C.3. The convolution operation ✶ is defined as d e 𝟤 𝑥𝑚 𝑦𝑛−𝑚 ∀⦅𝑥𝑛 ⦆𝑛∈ℤ ,⦅𝑦𝑛 ⦆𝑛∈ℤ ∈𝓵ℝ ⦅𝑥𝑛 ⦆ ✶ ⦅𝑦𝑛 ⦆ ≜ f ∑ ⦅𝑚∈ℤ ⦆ 𝑛∈ℤ

Definition C.4. 3 The z-transform 𝐙 of ⦅𝑥𝑛 ⦆𝑛∈ℤ is defined as −𝑛 𝟤 d ∀⦅𝑥𝑛 ⦆∈𝓵ℝ [𝐙 ⦅𝑥𝑛 ⦆](𝑧) ≜ ∑ 𝑥𝑛 𝑧 e 𝑛∈ℤ ⏟⏟⏟⏟⏟ f Laurent series

Proposition C.1. Let ✶ be the CONVOLUTION OPERATOR (Definition C.3 page 91). p 𝟤 r ⦅𝑥𝑛 ⦆ ✶ ⦅𝑦𝑛 ⦆ = ⦅𝑦𝑛 ⦆ ✶ ⦅𝑥𝑛 ⦆ ∀⦅𝑥𝑛 ⦆𝑛∈ℤ ,⦅𝑦𝑛 ⦆𝑛∈ℤ ∈𝓵ℝ (✶ is COMMUTATIVE) p 1

📘 Bromwich (1908), page 1, 📘 Thomson et al. (2008) page 23 ⟨Definition 2.1⟩, 📘 Joshi (1997) page 31 📘 Kubrusly (2011) page 347 ⟨Example 5.K⟩ 3 Laurent series: 📘 Abramovich and Aliprantis (2002) page 49 2

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Daniel J. Greenhoe

APPENDIX C. OPERATIONS ON SEQUENCES

✎PROOF: [𝑥✶𝑦](𝑛) ≜

∑

𝑥𝑚 𝑦𝑛−𝑚

by Definition C.3 page 91

∑

𝑥𝑛−𝑘 𝑦(𝑘)

where 𝑘 = 𝑛 − 𝑚 ⟺ 𝑚 = 𝑛 − 𝑘

∑

𝑥𝑛−𝑘 𝑦(𝑘)

by change commutivity of addition

∑

𝑥𝑛−𝑚 𝑦𝑚

by change of variables

∑

𝑦𝑚 𝑥𝑛−𝑚

by commutative property of the field over ℂ

𝑚∈ℤ

=

𝑘∈ℤ

=

𝑘∈ℤ

=

𝑚∈ℤ

=

𝑚∈ℤ

≜ (𝑦✶𝑥)𝑛


✏ Theorem C.1. Let ✶ be the convolution operator (Definition C.3 page 91). t 𝟤 h 𝐙 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ∀⦅𝑥𝑛 ⦆𝑛∈ℤ ,⦅𝑦𝑛 ⦆𝑛∈ℤ ∈𝓵ℝ (⦅𝑥𝑛 ⦆ ✶ ⦅𝑦𝑛 ⦆) = ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ (𝐙 ⦅𝑥𝑛 ⦆) (𝐙 ⦅𝑦𝑛 ⦆) m sequence convolution

series multiplication

✎PROOF: [𝐙(𝑥✶𝑦)](𝑧) ≜ 𝐙 ≜

∑ (𝑚∈ℤ

𝑥𝑚 𝑦𝑛−𝑚


)

∑∑

𝑥𝑚 𝑦𝑛−𝑚 𝑧−𝑛

∑∑


∑∑


∑∑

𝑥𝑚 𝑦𝑘 𝑧−(𝑚+𝑘)



=


=

𝑚∈ℤ 𝑛∈ℤ

=

where 𝑘 = 𝑛 − 𝑚 ⟺ 𝑛 = 𝑚 + 𝑘

𝑚∈ℤ 𝑘∈ℤ

=

[∑

𝑥𝑚 𝑧−𝑚

𝑚∈ℤ

][ ∑

𝑦𝑘 𝑧−𝑘

𝑘∈ℤ

] by Definition C.4 page 91

≜ (𝐙 ⦅𝑥𝑛 ⦆) (𝐙 ⦅𝑦𝑛 ⦆)

✏

C.2

Zero Locations

The system property of minimum phase is defined in Definition C.5 (next) and illustrated in Figure C.1 (page 93). ̂ Definition C.5. 4 Let 𝗑(𝑧) ≜ 𝐙 ⦅𝑥𝑛 ⦆ be the Z TRANSFORM (Definition C.4 page 91) of a sequence ⦅𝑥𝑛 ⦆𝑛∈ℤ in 𝟤 ̂ 𝓵ℝ . Let ⦅𝑧𝑛 ⦆𝑛∈ℤ be the ZEROS of 𝗑(𝑧). The sequence ⦅𝑥𝑛 ⦆ is minimum phase if d 𝑧𝑛 | < 1 ∀𝑛 ∈ ℤ |⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ e f ̂ has all its ZEROS inside the unit circle 𝗑(𝑧) 4

📘 Farina and Rinaldi (2000) page 91, 📘 Dumitrescu (2007) page 36

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Daniel J. Greenhoe

page 93

ℑ [𝑧]

×

c c

×

c

×

c

ℜ [𝑧]

×

Figure C.1: Minimum Phase filter

The impulse response of a minimum phase filter has most of its energy concentrated near the beginning of its support, as demonstrated next. Theorem C.2 (Robinson's Energy Delay Theorem). be polynomials. t 𝗉 is MINIMUM PHASE h m { 𝗊 is NOT minimum phase

5

𝑚−1

and

2

𝑚−1

2

≥ 𝑏 𝑎 ∑ | 𝑛| ∑ | 𝑛| 𝑛=0 𝑛=0 ⏟⏟⏟ ⏟⏟⏟

⟹

}

−𝑛 −𝑛 Let 𝗉(𝑧) ≜ ∑𝘕 and 𝗊(𝑧) ≜ ∑𝘕 𝑛=0 𝑎𝑛 𝑧 𝑛=0 𝑏𝑛 𝑧

“energy” of the first 𝑚 coefficients of 𝗉(𝑧)

∀0 ≤ 𝑚 ≤ 𝘕

“energy” of the first 𝑚 coefficients of 𝗊(𝑧)

Example C.1. An example of a minimum phase polynomial is the Daubechies-4 scaling function. The minimum phase polynomial causes most of the energy to be concentrated near the origin, making it very asymmetric. In contrast, the Symlet-4 has a design very similar to that of Daubechies4, but the selected zeros are not all within the unit circle in the complex 𝑧 plane. This results in a scaling function that is more symmetric and less contrated near the origin. Both scaling functions are illustrated in Figure C.2 (page 94).

C.3 Rational polynomial operators 𝟤 . If the digital filter is a causal LTI system, then it can be A digital filter is simply an operator on 𝓵ℝ expressed as a rational polynomial in 𝑧 as shown next.

Lemma C.1. A causal LTI operator ℋ can be expressed as a rational expression 𝐻(𝑧). 𝐻(𝑧) =

𝑏0 + 𝑏1 𝑧−1 + 𝑏2 𝑧−2 + ⋯ + 𝑏𝑁 𝑧−𝑁 1 + 𝑎1 𝑧−1 + 𝑎2 𝑧−2 + ⋯ + 𝑎𝑁 𝑧−𝑁 𝑁

∑ 𝑏𝑛 𝑧−𝑛

=

𝑛=0

𝑁

1 + ∑ 𝑎𝑛 𝑧−𝑛 𝑛=1

5

📘 Dumitrescu (2007) page 36, 📘 Robinson (1962), 📘 Robinson (1966) ⟨???⟩, 📘 Claerbout (1976), pages 52–53



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𝑝 − 1 = 3 discarded zeroes

Daubechies-4 2𝑝−1 = 7 poles ℑ[𝑧]

7

4 |𝑧|=1

𝑝= 4 zeroes

Symlets-4 2𝑝−1 = 7 poles ℑ[𝑧]

7

4

ℜ[𝑧]

ℜ[𝑧]

|𝑧|=1

𝑝= 4 zeroes symlet 𝜙(𝑡) and ⦅ℎ𝑛 ⦆ for 𝑝 = 4

1 0

1

3 4

7

5 0

1 2

5 6

1

−1

2

3

4

6

7

−1

minimum phase

not minimum phase

Figure C.2: Daubechies-4 and Symlet-4 scaling functions pole-zero plots

A filter operation 𝐻(𝑧) can be expressed as a product of its roots (poles and zeros). 𝐻(𝑧) =

𝑏0 + 𝑏1 𝑧−1 + 𝑏2 𝑧−2 + ⋯ + 𝑏𝘕 𝑧−𝘕

1 + 𝑎1 𝑧−1 + 𝑎2 𝑧−2 + ⋯ + 𝑎𝘕 𝑧−𝘕 (𝑧 − 𝑧1 )(𝑧 − 𝑧2 ) ⋯ (𝑧 − 𝑧𝘕 ) = 𝛼 (𝑧 − 𝑝1 )(𝑧 − 𝑝2 ) ⋯ (𝑧 − 𝑝𝘕 )

where 𝛼 is a constant, 𝑧𝑖 are the zeros, and 𝑝𝑖 are the poles. The poles and zeros of such a rational expression are often plotted in the z-plane with a unit circle about the origin (representing 𝑧 = 𝑒𝑖𝜔 ). Poles are marked with an × and zeros with an ○. An example is shown in Figure C.3 page 94. Notice that in this figure the zeros and poles are either real or occur in complex conjugate pairs. ℑ [𝑧] c × c

× ×

ℜ [𝑧] c

Figure C.3: Pole-zero plot for rational expression with real coefficients

Theorem C.3. A causal LTI filter is stable if all of its poles are inside the unit circle.

Stable/unstable filters are illustrated in Figure C.4 (page 95). CC

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C.4. SAMPLE RATE CONVERSION

Daniel J. Greenhoe ℑ [𝑧]

ℑ [𝑧]

×

c

c

×

c

c

×

×

ℜ [𝑧]

×

c

page 95

c ×

ℜ [𝑧]

×

c

stable

c unstable

×

Figure C.4: Pole-zero plot stable/unstable causal LTI filters

C.4 Sample rate conversion Theorem C.4 (upsampling). Let ⦅𝑥𝑛 ⦆𝑛∈ℤ and ⦅𝑦𝑛 ⦆𝑛∈ℤ be SEQUENCES (Definition C.1 page 91) in 𝓵𝔽𝟤 C.2 page 91) over a FIELD 𝔽 . t 𝑥(𝑛/𝘕 ) for 𝑛 mod 𝘕 = 0 h 𝗒(𝑛) = ̂ = 𝗑̂(𝑧𝘕 ) ⟹ 𝗒(𝑧) { } 0 otherwise m

(Definition

✎PROOF: ̂ ≜ 𝗒(𝑧)

𝑦𝑛 𝑧−𝑛

∑

by definition of z-transform (Definition C.4 page 91)

𝑛∈ℤ

=

∑

𝑦𝑛 𝑧−𝑛 +

𝑛 mod 𝘕 =0

=

∑

𝑛

𝑥𝑛/𝘕 𝑧

𝑛 mod 𝘕 =0

=

∑

𝑥𝑚 𝑧−𝑚𝘕

∑

𝑥𝑚 (𝑧𝘕 )

−𝑛

+

∑


mod 𝘕 ≠0

:0 −𝑛 0𝑧 ∑ 𝑛 mod 𝘕 ≠0

by definition of ⦅𝑦𝑛 ⦆ where 𝑚 ≜ 𝑛/𝘕 ⟹ 𝑛 = 𝑚𝘕

𝑚∈ℤ

=

−𝑚

𝑚∈ℤ 𝘕

≜ 𝗑̂(𝑧 )


✏ Theorem C.5 (downsampling). Let ⦅𝑥𝑛 ⦆𝑛∈ℤ and ⦅𝑦𝑛 ⦆𝑛∈ℤ be SEQUENCES (Definition C.1 page 91) in 𝓵𝔽𝟤 nition C.2 page 91) over a FIELD 𝔽 . 𝘕 −1 ⎧ ⎫ 2𝜋𝑚 1 ⎪ t ⎪ h {𝑦𝑛 = 𝑥(𝘕 𝑛 )} ̂ = 𝗑̂ (𝑒𝑖 𝘕 𝑧 𝘕 )⎬ 𝗒(𝑧) ⟹ ⎨ ∑ m ⎪ ⎪ 𝑚=0 ⎩ ⎭

(Defi-

✎PROOF: ̂ ≜ 𝗒(𝑧)

∑



∑

𝑥(𝑛𝘕 ) 𝑧−𝑛

by definition of ⦅𝑦𝑛 ⦆

∑

𝑥𝑛 [𝛿(𝑛̄

𝑛∈ℤ

=

𝑛∈ℤ

=

𝑛∈ℤ

− 𝘕𝑛 mod 𝘕 ) ]𝑧



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=

⎡𝘕 −1 −𝑖 2𝜋𝑛𝑚 ⎤ − 𝑛 𝑥𝑛 ⎢ 𝑒 𝘕 ⎥𝑧 𝘕 ∑ ⎢∑ ⎥ 𝑛∈ℤ 𝑚=0 ⎣ ⎦ 𝘕 −1

=

∑∑

𝑚=0 𝑛∈ℤ

𝑥𝑛 (𝑒𝑖

2𝜋𝑚 𝘕

𝑥𝑛 (𝑒𝑖

2𝜋𝑚 𝘕

𝘕 −1

=

∑∑

𝑚=0 𝑛∈ℤ 𝘕 −1

≜


𝗑̂ 𝑒𝑖 ∑ (

𝑚=0

2𝜋𝑚 𝘕

−𝑛

1

𝘕 ) (𝑧 )

1

𝑧𝘕 )

−𝑛

−𝑛

1

𝑧𝘕 )


✏

C.5

Filter Banks

̂ is a low-pass filter, then Conjugate quadrature filters (next definition) are used in filter banks. If 𝗑(𝑧) ̂ is a high-pass filter. the conjugate quadrature filter of 𝗒(𝑧) 𝟤 Definition C.6. 6 Let ⦅𝑥𝑛 ⦆𝑛∈ℤ and ⦅𝑦𝑛 ⦆𝑛∈ℤ be SEQUENCES (Definition C.1 page 91) in 𝓵ℝ (Definition C.2 page 91). The sequence ⦅𝑦𝑛 ⦆ is a conjugate quadrature filter with shift 𝘕 with respect to ⦅𝑥𝑛 ⦆ if 𝑦𝑛 = ±(−1)𝑛 𝑥∗𝘕 −𝑛 d e A CONJUGATE QUADRATURE FILTER is also called a CQF or a Smith-Barnwell filter. f Any triple ⦅𝑥 ⦆ , ⦅𝑦 ⦆ , 𝘕 in this form is said to satisfy the ( 𝑛 ) 𝑛 conjugate quadrature filter condition or the CQF condition.

̆ and 𝗑(𝜔) ̆ Theorem C.6. 7 Let 𝗒(𝜔) be the DTFTs (Definition 6.1 page 69) of the sequences ⦅𝑦𝑛 ⦆𝑛∈ℤ and ⦅𝑥𝑛 ⦆𝑛∈ℤ , 𝟤 respectively, in 𝓵ℝ (Definition C.2 page 91). −1 𝑛 ∗ ̂ 𝑦⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⟺ 𝗒(𝑧) = ±(−1)𝘕 𝑧−𝘕 𝗑̂∗ ( ∗ ) (2) CQF in “z-domain” 𝑛 = ±(−1) 𝑥𝘕 −𝑛 𝑧 (1) CQF in “time”

̆ ⟺ 𝗒(𝜔) = ±(−1)𝘕 𝑒−𝑖𝜔𝘕 𝗑̆∗ (𝜔 + 𝜋) ⟺ 𝑥𝑛 = ±(−1)𝘕 (−1)𝑛 𝑦∗𝘕 −𝑛 −1 ̂ ⟺ 𝗑(𝑧) = ±𝑧−𝘕 𝗒∗̂ ( ∗ ) 𝑧 ̆ ⟺ 𝗑(𝜔) = ±𝑒−𝑖𝜔𝘕 𝗒∗̆ (𝜔 + 𝜋)

t h m

(3)

CQF in “frequency”

(4)

“reversed” CQF in “time”

(5)

“reversed” CQF in “z-domain”

(6)

“reversed” CQF in “frequency”

∀𝘕 ∈ℤ

✎PROOF: 6

📘 Strang and Nguyen (1996) page 109, 📘 Haddad and Akansu (1992) pages 256–259 ⟨section 4.5⟩, 📘 Vaidyanathan (1993) page 342 ⟨(7.2.7), (7.2.8)⟩, 📘 Smith and Barnwell (1984a), 📘 Smith and Barnwell (1984b), 📘 Mintzer (1985) 7 📘 Strang and Nguyen (1996) page 109, 📘 Mallat (1999) pages 236–238 ⟨(7.58),(7.73)⟩, 📘 Haddad and Akansu (1992) pages 256–259 ⟨section 4.5⟩, 📘 Vaidyanathan (1993) page 342 ⟨(7.2.7), (7.2.8)⟩ CC

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1. Proof that (1) ⟹ (2): ̂ = 𝗒(𝑧)

∑



𝑛∈ℤ

=

(±)(−1)𝑛 𝑥∗𝘕 −𝑛 𝑧−𝑛 ∑ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

by (1)

𝑛∈ℤ

CQF

=±

∑

(−1)𝘕 −𝑚 𝑥∗𝑚 𝑧−(𝘕 −𝑚)

where 𝑚 ≜ 𝘕 − 𝑛 ⟹ 𝑛 = 𝘕 − 𝑚

𝑚∈ℤ

= ±(−1)𝘕 𝑧−𝘕

∑

(−1)−𝑚 𝑥∗𝑚 (𝑧−1 )

−𝑚

𝑚∈ℤ

1 −𝑚 𝑥∗𝑚 (− ) ∑ 𝑧 𝑚∈ℤ

= ±(−1)𝘕 𝑧−𝘕

∗

𝘕 −𝘕

= ±(−1) 𝑧

1 −𝑚 𝑥𝑚 (− ∗ ) ∑ 𝑧 [𝑚∈ℤ ]

= ±(−1)𝘕 𝑧−𝘕 𝗑̂∗ (

−1 𝑧∗ )


2. Proof that (1) ⟸ (2): ̂ = ±(−1)𝘕 𝑧−𝘕 𝗑̂∗ ( 𝗒(𝑧) = ±(−1)𝘕 𝑧−𝘕 = ±(−1)𝘕 𝑧−𝘕 =

−1 𝑧∗ )

∑ [𝑚∈ℤ ∑ [𝑚∈ℤ

𝑥𝑚 (

by (2) −1 −𝑚 𝑧∗ ) ]

𝑥∗𝑚 (−𝑧−1 )

∑

(±)(−1)𝘕 −𝑚 𝑥∗𝑚 𝑧−(𝘕 −𝑚)

∑

(±)(−1)𝑛 𝑥∗𝘕 −𝑛 𝑧−𝑛

∗

−𝑚

]

by definition of z-transform (Definition C.4 page 91) by definition of z-transform (Definition C.4 page 91)

𝑚∈ℤ

=

where 𝑛 = 𝘕 − 𝑚 ⟹ 𝑚 ≜ 𝘕 − 𝑛

𝑚∈ℤ

⟹ 𝑥𝑛 = ±(−1)𝑛 𝑥∗𝘕 −𝑛 3. Proof that (1) ⟹ (3): ̆ ̂ | 𝗒(𝜔) ≜ 𝗑(𝑧)


𝑧=𝑒𝑖𝜔

−1 𝑧∗ )]𝑧=𝑒𝑖𝜔 𝘕 −𝑖𝜔𝘕 = ±(−1) 𝑒 𝗑̂(𝑒𝑖𝜋 𝑒𝚤𝜔 )

= [±(−1)𝘕 𝑧−𝘕 𝗑̂(


by (2)

= ±(−1)𝘕 𝑒−𝑖𝜔𝘕 𝗑̂(𝑒𝑖(𝜔+𝜋) ) ̆ + 𝜋) = ±(−1)𝘕 𝑒−𝑖𝜔𝘕 𝗑(𝜔



4. Proof that (1) ⟹ (6): ̆ 𝗑(𝜔) =

∑

𝑦𝑛 𝑒−𝑖𝜔𝑛



𝑛∈ℤ

=

±(−1)𝑛 𝑥∗𝘕 −𝑛 𝑒−𝑖𝜔𝑛 ∑ ⏟⏟⏟⏟⏟⏟⏟⏟⏟

by (1)

𝑛∈ℤ

CQF

=

∑

±(−1)𝘕 −𝑚 𝑥∗𝑚 𝑒−𝑖𝜔(𝘕 −𝑚)


𝑚∈ℤ

= ±(−1)𝘕 𝑒−𝑖𝜔𝘕

∑

(−1)𝑚 𝑥∗𝑚 𝑒𝑖𝜔𝑚

𝑚∈ℤ Tuesday 3rd October, 2017 3:01am UTC Copyright ©2017 Daniel J. Greenhoe


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Daniel J. Greenhoe

= ±(−1)𝘕 𝑒−𝑖𝜔𝘕


∑

𝑒𝑖𝜋𝑚 𝑥∗𝑚 𝑒𝑖𝜔𝑚

∑

𝑥∗𝑚 𝑒𝑖(𝜔+𝜋)𝑚

𝑚∈ℤ 𝘕 −𝑖𝜔𝘕

= ±(−1) 𝑒

𝑚∈ℤ ∗ 𝘕 −𝑖𝜔𝘕

= ±(−1) 𝑒

∑ [𝑚∈ℤ

𝑥𝑚 𝑒

−𝑖(𝜔+𝜋)𝑚

]

= ±(−1)𝘕 𝑒−𝑖𝜔𝘕 𝗑̆∗ (𝜔 + 𝜋)



5. Proof that (1) ⟸ (3): +𝜋

𝑦𝑛 =

1 𝑖𝜔𝑛 ̆ 𝗒(𝜔)𝑒 𝖽𝜔 2𝜋 ∫ −𝜋


+𝜋

=

1 𝘕 −𝑖𝘕 𝜔 ∗ ±(−1) 𝑒 𝗑̆ (𝜔 + 𝜋)𝑒𝑖𝜔𝑛 𝖽𝜔 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 2𝜋 ∫ −𝜋

by right hypothesis

right hypothesis +𝜋

= ±(−1)𝘕

1 𝗑̆∗ (𝜔 + 𝜋)𝑒𝑖𝜔(𝑛−𝘕 ) 𝖽𝜔 2𝜋 ∫ −𝜋

= ±(−1)𝘕

1 2𝜋 ∫ 0

2𝜋

= ±(−1)𝘕 𝑒−𝑖𝜋(𝑛−𝘕 )

𝗑̆∗ (𝑣)𝑒𝑖(𝑣−𝜋)(𝑛−𝘕 ) 𝖽𝑣 1 2𝜋 ∫ 0

2𝜋

𝘕

𝑛

where 𝑣 ≜ 𝜔 + 𝜋 ⟹ 𝜔 = 𝑣 − 𝜋

𝗑̆∗ (𝑣)𝑒𝑖𝑣(𝑛−𝘕 ) 𝖽𝑣

1 = ±(−1) (−1) ⏟ (−1) ⏟[ ∫ 2𝜋 0 −𝑖𝜋𝑛 𝑖𝜋𝑁 𝘕

by right hypothesis

∗

2𝜋

𝑖𝑣(𝘕 −𝑛)

̆ 𝗑(𝑣)𝑒

𝖽𝑣 ]

𝑒

𝑒

= ±(−1)𝑛 𝑥∗𝘕 −𝑛


6. Proof that (1) ⟺ (4): 𝑦𝑛 = ±(−1)𝑛 𝑥∗𝘕 −𝑛 ⟺ (±)(−1)𝑛 𝑦𝑛 = (±)(±)(−1)𝑛 (−1)𝑛 𝑥∗𝘕 −𝑛 ⟺ ±(−1)𝑛 𝑦𝑛 = 𝑥∗𝘕 −𝑛 ∗

∗

⟺ (±(−1)𝑛 𝑦𝑛 ) = (𝑥∗𝘕 −𝑛 ) ⟺ ±(−1)𝑛 𝑦∗𝑛 = 𝑥𝘕 −𝑛 ⟺ 𝑥𝘕 −𝑛 = ±(−1)𝑛 𝑦∗𝑛 ⟺ 𝑥𝑚 = ±(−1)𝘕 −𝑚 𝑦∗𝘕 −𝑚 ⟺ ⟺ ⟺


𝑥𝑚 = ±(−1)𝘕 −𝑚 𝑦∗𝘕 −𝑚 𝑥𝑚 = ±(−1)𝘕 (−1)𝑚 𝑦∗𝘕 −𝑚 𝑥𝑛 = ±(−1)𝘕 (−1)𝑛 𝑦∗𝘕 −𝑛

by change of free variables

7. Proofs for (5) and (6): not included. See proofs for (2) and (3).

✏ ̆ and 𝗑(𝜔) ̆ Theorem C.7. 8 Let 𝗒(𝜔) be the DTFTs (Definition 6.1 page 69) of the sequences ⦅𝑦𝑛 ⦆𝑛∈ℤ and ⦅𝑥𝑛 ⦆𝑛∈ℤ , 𝟤 respectively, in 𝓵ℝ (Definition C.2 page 91). Let 𝑦𝑛 = ±(−1)𝑛 𝑥∗𝑁−𝑛 ( CQF CONDITION, Definition C.6 page 96). Then ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

t h m

8

(A)

𝗱 [ 𝗱𝞈 ]

𝑛

̆ | 𝗒(𝜔)

𝜔=0

=0

𝑛

𝗱 ̆ | ⟺ [ 𝗱𝞈 = 0 ] 𝗑(𝜔) 𝜔=𝜋 ⟺ (−1)𝑘 𝑘𝑛 𝑥𝑘 = 0 ∑

(B) (C)

𝑘∈ℤ

⟺

∑

𝑛

𝑘 𝑦𝑘

= 0

𝑘∈ℤ

(D)

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

∀𝑛∈𝕎

📘 Vidakovic (1999), pages 82–83, 📘 Mallat (1999), pages 241–242

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Daniel J. Greenhoe

page 99

✎PROOF: 1. Proof that (A) ⟹ (B): 𝑛

𝗱 ̆ | 0 = [ 𝗱𝞈 ] 𝗒(𝜔) 𝜔=0

by (A)

𝑛

𝗱 𝘕 −𝑖𝜔𝘕 ∗ = [ 𝗱𝞈 𝗑̆ (𝜔 + 𝜋)| ] (±)(−1) 𝑒

by Theorem C.6 page 96

𝜔=0

𝑛

= (±)(−1)

𝘕

𝑛−ℓ 𝑛 𝗱 ℓ −𝑖𝜔𝘕 𝗱 ∗ 𝑒 ⋅ [ ] [ 𝗱𝞈 ] [𝗑̆ (𝜔 + 𝜋)]| ∑ (ℓ)[ 𝗱𝞈 ] ℓ=0

by Leibnitz GPR (Lemma B.2 page 89) 𝜔=0

𝑛

= (±)(−1)𝘕

𝑛−ℓ 𝑛 𝗱 ∗ −𝑖𝘕 ℓ 𝑒−𝑖𝜔𝘕 [ 𝗱𝞈 [𝗑̆ (𝜔 + 𝜋)] ( ) ] ∑ ℓ | ℓ=0

𝑛 𝑛−ℓ *1 𝑛 𝘕 −𝑖0𝘕 ℓ 𝗱 ∗ 𝑒 −𝑖𝘕 = (±)(−1) [ 𝗱𝞈 ] [𝗑̆ (𝜔 + 𝜋)]| ∑ (ℓ) ℓ=0

𝗑̆(0) (𝜋) 𝗑̆(1) (𝜋) 𝗑̆(2) (𝜋) 𝗑̆(3) (𝜋) 𝗑̆(4) (𝜋) ⋮ 𝗑̆(𝑛) (𝜋)

⟹ ⟹ ⟹ ⟹ ⟹ ⋮ ⟹

= = = = =

𝜔=0

𝜔=0

0 0 0 0 0

= 0 for 𝑛 = 0, 1, 2, …

2. Proof that (A) ⟸ (B): 𝑛

𝗱 ̆ | 0 = [ 𝗱𝞈 ] 𝗑(𝜔) 𝜔=𝜋

by (B)

𝑛

𝗱 −𝑖𝜔𝘕 ∗ = [ 𝗱𝞈 𝗒̆ (𝜔 + 𝜋)| ] (±)𝑒

by Theorem C.6 page 96

𝜔=𝜋

𝑛

= (±)

𝑛−ℓ 𝑛 𝗱 ℓ −𝑖𝜔𝘕 𝗱 ∗ 𝑒 ⋅ [ [𝗒̆ (𝜔 + 𝜋)] ] ( )[ ] ] [ 𝗱𝞈 ∑ ℓ 𝗱𝞈 | ℓ=0 𝑛

= (±)

𝑛−ℓ

𝑛 𝗱 (−𝑖𝘕 )ℓ 𝑒−𝑖𝜔𝘕 [ 𝗱𝞈 ] ∑ (ℓ) ℓ=0

𝑛

:1

= (±) 𝑒−𝑖𝜋𝘕

𝑛−ℓ

𝑛 𝗱 −𝑖𝘕 ℓ [ 𝗱𝞈 ] ∑ (ℓ)

∗ [𝗒̆ (𝜔 + 𝜋)] |

∗ [𝗒̆ (𝜔 + 𝜋)]

ℓ=0

|

by Leibnitz GPR (Lemma B.2 page 89) 𝜔=𝜋

𝜔=𝜋

𝜔=𝜋

𝑛

= (±)(−1)𝘕

𝑛−ℓ 𝑛 𝗱 ∗ −𝑖𝘕 ℓ [ 𝗱𝞈 [𝗒̆ (𝜔 + 𝜋)] ( ) ] ∑ ℓ | ℓ=0

⟹ ⟹ ⟹ ⟹ ⟹ ⋮ ⟹ ⟹

̆ (0) 𝗒(0) ̆ (0) 𝗒(1) (2) 𝗒̆ (0) ̆ (0) 𝗒(3) (4) 𝗒̆ (0) ⋮ ̆ (0) 𝗒(𝑛) ̆ (0) 𝗒(𝑛)

= = = = =

𝜔=𝜋

0 0 0 0 0

= 0 = 0 for 𝑛 = 0, 1, 2, …

3. Proof that (B) ⟺ (C): by Theorem 6.3 page 75 4. Proof that (A) ⟺ (D): by Theorem 6.3 page 75 Tuesday 3rd October, 2017 3:01am UTC Copyright ©2017 Daniel J. Greenhoe


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̆ 5. Proof that (CQF) ⟸ = 0. / (A): Here is a counterexample: 𝗒(𝜔)

✏

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APPENDIX D OPERATORS ON LINEAR SPACES

❝And I

am not afraid to say that there is a way to advance algebra as far beyond what Vieta and Descartes have left us as Vieta and Descartes carried it beyond the ancients.…we need still another analysis which is distinctly geometrical or linear, and which will express situation directly as algebra expresses magnitude directly.❞ Gottfried Leibniz (1646–1716), German mathematician, in a September 8, 1679 letter to Christian Huygens. 1

D.1 Operators on linear spaces D.1.1 Operator Algebra An operator is simply a function that maps from a linear space to another linear space (or to the same linear space).

Definition D.1. 2 Let (𝔽 , +, ⋅, 0, 1) be a FIELD. Let 𝑋 be a set, let + be an OPERATOR (Definition D.3 page 102) 2 in 𝑋 𝑋 , and let ⊗ be an operator in 𝑋 𝔽 ×𝑋 .

1

quote: 📘 Leibniz (1679) pages 248–249 image: http://en.wikipedia.org/wiki/File:Gottfried_Wilhelm_von_Leibniz.jpg, public domain 2 📘 Kubrusly (2001) pages 40–41 ⟨Definition 2.1 and following remarks⟩, 📘 Haaser and Sullivan (1991), page 41, 📘 Halmos (1948), pages 1–2, 📘 Peano (1888a) ⟨Chapter IX⟩, 📘 Peano (1888b), pages 119–120, 📃 Banach (1922) pages 134–135

page 102

Daniel J. Greenhoe

APPENDIX D. OPERATORS ON LINEAR SPACES relations

functions

operators (Definition D.3 page 102)

linear operators (Definition D.5 page 103)

non-linear operators

normal operators 𝐍∗ 𝐍 = 𝐍𝐍∗

projection operators 𝐏2 = 𝐏

isometric operators 𝐀∗ 𝐀 = 𝐈




self adjoint operators 𝐀 = 𝐀∗

unitary operators 𝐔𝐔∗ = 𝐔∗ 𝐔 = 𝐈



Figure D.1: Some operator types

The structure 𝞨 ≜ (𝑋, +, ⋅, (𝔽 , ∔, ⨰)) is a linear space over (𝔽 , +, ⋅, 0, 1) if 1. ∃𝟘 ∈ 𝑋 such that 𝒙 + 𝟘 = 𝒙 ∀𝒙∈𝑋 (+ IDENTITY) ∗ 2. ∃𝒚 ∈ 𝑋 such that 𝒙 + 𝒚 = 𝟘 ∀𝒙∈𝑋 (+ INVERSE) 3. (𝒙 + 𝒚) + 𝒛 = 𝒙 + (𝒚 + 𝒛) ∀𝒙,𝒚,𝒛∈𝑋 (+ is ASSOCIATIVE) 4. 𝒙+𝒚 = 𝒚+𝒙 ∀𝒙,𝒚∈𝑋 (+ is COMMUTATIVE) 5. 1⋅𝒙 = 𝒙 ∀𝒙∈𝑋 (⋅ IDENTITY) 6. 𝛼 ⋅ (𝛽 ⋅ 𝒙) = (𝛼 ⋅ 𝛽) ⋅ 𝒙 ∀𝛼,𝛽∈𝑆 and 𝒙∈𝑋 (⋅ ASSOCIATES with ⋅) 7. 𝛼 ⋅ (𝒙 + 𝒚) = (𝛼 ⋅ 𝒙) + (𝛼 ⋅ 𝒚) ∀𝛼∈𝑆 and 𝒙,𝒚∈𝑋 (⋅ DISTRIBUTES over +) 8. (𝛼 + 𝛽) ⋅ 𝒙 = (𝛼 ⋅ 𝒙) + (𝛽 ⋅ 𝒙) ∀𝛼,𝛽∈𝑆 and 𝒙∈𝑋 (⋅ PSEUDO-DISTRIBUTES over +) The set 𝑋 is called the underlying set. The elements of 𝑋 are called vectors. The elements of 𝔽 are called scalars. A linear space is also called a vector space. If 𝔽 ≜ ℝ, then 𝞨 is a real linear space. If 𝔽 ≜ ℂ, then 𝞨 is a complex linear space.

d e f

Definition D.2. d e The Kronecker delta function 𝛿𝑛̄ is defined as f

𝛿𝑛̄ ≜

1 for 𝑛 = 0 { 0 for 𝑛 ≠ 0.̇

and

∀𝑛∈ℤ

Definition D.3. 3 d A function 𝐀 in 𝙔 𝙓 is an operator in 𝙔 𝙓 if e f 𝙓 and 𝙔 are both LINEAR SPACES (Definition D.1 page 101). Two operators 𝐀 and 𝐁 in 𝙔 𝙓 are equal if 𝐀𝒙 = 𝐁𝒙 for all 𝒙 ∈ 𝙓 . The inverse relation of an operator 𝐀 in 𝙔 𝙓 always exists as a relation in 𝟚𝙓 𝙔 , but may not always be a function (may not always be an operator) in 𝙔 𝙓 . The operator 𝐈 ∈ 𝙓 𝙓 is the identity operator if 𝐈𝒙 = 𝐈 for all 𝒙 ∈ 𝙓 . Definition D.4. 3 4

4

Let 𝙓 𝙓 be the set of all operators with from a linear space 𝙓 to 𝙓 . Let 𝐈 be an

📘 Heil (2011) page 42 📘 Michel and Herget (1993) page 411

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operator in 𝙓 𝙓 . Let ○(𝙓 ) be the IDENTITY ELEMENT in 𝙓 𝙓 . d e 𝐈 is the identity operator in 𝙓 𝙓 if 𝐈 = ○(𝙓 ). f

D.1.2 Linear operators Definition D.5.

5

Let 𝙓 ≜ (𝑋, +, ⋅, (𝔽 , ∔, ⨰)) and 𝙔 ≜ (𝑌 , +, ⋅, (𝔽 , ∔, ⨰)) be linear spaces.

An operator 𝐋 ∈ 𝙔 𝙓 is linear if 1. 𝐋(𝒙 + 𝒚) = 𝐋𝒙 + 𝐋𝒚 ∀𝒙,𝒚∈𝙓 ( ADDITIVE) and 2. 𝐋(𝛼𝒙) = 𝛼𝐋𝒙 ∀𝒙∈𝙓 , ∀𝛼∈𝔽 ( HOMOGENEOUS). The set of all linear operators from 𝙓 to 𝙔 is denoted ℒ(𝙓 , 𝙔 ) such that ℒ(𝙓 , 𝙔 ) ≜ {𝐋 ∈ 𝙔 𝙓 |𝐋 is linear} .

d e f

Theorem D.1.

6

Let 𝐋 be an operator from a linear space 𝙓 to a linear space 𝙔 , both over a field 𝔽 . 1. 𝐋𝟘 = 𝟘 ⎧ ⎪ 2. 𝐋(−𝒙) = −(𝐋𝒙) ∀𝒙∈𝙓 ⎪ t 3. 𝐋(𝒙 − 𝒚) = 𝐋𝒙 − 𝐋𝒚 ∀𝒙,𝒚∈𝙓 ⎪ h 𝐋 is LINEAR ⟹ ⎨ 𝑁 m ⎞ ⎛𝑁 ⎪ ⎟ ⎜ ⎪ 4. 𝐋 ∑ 𝛼𝑛 𝒙𝑛 = ∑ 𝛼𝑛 (𝐋𝒙𝑛 ) 𝒙𝑛 ∈𝙓 , 𝛼𝑛 ∈𝔽 ⎟ ⎜𝑛=1 ⎪ 𝑛=1 ⎩ ⎠ ⎝

✎PROOF: 1. Proof that 𝐋𝟘 = 𝟘: 𝐋𝟘 = 𝐋(0 ⋅ 𝟘)

by additive identity property

= 0 ⋅ (𝐋𝟘)

by homogeneous property of 𝐋 (Definition D.5 page 103)

=𝟘

by additive identity property

2. Proof that 𝐋(−𝒙) = −(𝐋𝒙): 𝐋(−𝒙) = 𝐋(−1 ⋅ 𝒙)

by additive inverse property

= −1 ⋅ (𝐋𝒙)

by homogeneous property of 𝐋 (Definition D.5 page 103)

= −(𝐋𝒙)


3. Proof that 𝐋(𝒙 − 𝒚) = 𝐋𝒙 − 𝐋𝒚: 𝐋(𝒙 − 𝒚) = 𝐋(𝒙 + (−𝒚))


= 𝐋(𝒙) + 𝐋(−𝒚)

by linearity property of 𝐋 (Definition D.5 page 103)

= 𝐋𝒙 − 𝐋𝒚

by 2.

𝑁 4. Proof that 𝐋(∑𝑁 𝑛=1 𝛼𝑛 𝒙𝑛 ) = ∑𝑛=1 𝛼𝑛 (𝐋𝒙𝑛 ): 5

📘 Kubrusly (2001) page 55, 📘 Aliprantis and Burkinshaw (1998) page 224, 📘 Hilbert et al. (1927) page 6, 📘 Stone (1932) page 33 6 📘 Berberian (1961) page 79 ⟨Theorem IV.1.1⟩ Tuesday 3rd October, 2017 3:01am UTC Copyright ©2017 Daniel J. Greenhoe


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(a) Proof for 𝑁 = 1: 𝑁

𝐋 𝛼𝑛 𝒙𝑛 = 𝐋(𝛼1 𝒙1 ) (∑ ) 𝑛=1

by 𝑁 = 1 hypothesis by homogeneous property of Definition D.5 page 103

= 𝛼1 (𝐋𝒙1 ) (b) Proof that 𝑁 case ⟹ 𝑁 + 1 case:

𝑁 ⎛𝑁+1 ⎞ ⎜ ⎟ 𝐋 𝛼𝑛 𝒙𝑛 = 𝐋 𝛼𝑁+1 𝒙𝑁+1 + 𝛼 𝒙 ∑ 𝑛 𝑛) ⎜∑ ⎟ ( 𝑛=1 𝑛=1 ⎝ ⎠ 𝑁

= 𝐋(𝛼𝑁+1 𝒙𝑁+1 ) + 𝐋

(∑ 𝑛=1

𝛼𝑛 𝒙𝑛

)

by linearity property of Definition D.5 page 103

𝑁

= 𝛼𝑁+1 𝐋(𝒙𝑁+1 ) +

𝐋 𝛼 𝒙 ∑ ( 𝑛 𝑛)

by left 𝑁 + 1 hypothesis

𝑛=1

𝑁+1

=

𝐋 𝛼 𝒙 ∑ ( 𝑛 𝑛) 𝑛=1

✏ Theorem D.2. 7 Let ℒ(𝙓 , 𝙔 ) be the set of all linear operators from a linear space 𝙓 to a linear space 𝙔 . Let 𝓝 (𝐋) be the NULL SPACE of an operator 𝐋 in 𝙔 𝙓 and 𝓘(𝐋) the IMAGE SET of 𝐋 in 𝙔 𝙓 . (space of linear transforms) t ℒ(𝙓 , 𝙔 ) is a linear space 𝙓 h 𝓝 (𝐋) is a linear subspace of 𝙓 ∀𝐋∈𝙔 m 𝓘(𝐋) is a linear subspace of 𝙔 ∀𝐋∈𝙔 𝙓 ✎PROOF: 1. Proof that 𝓝 (𝐋) is a linear subspace of 𝙓 : (a) 𝟘 ∈ 𝓝 (𝐋) ⟹ 𝓝 (𝐋) ≠ ∅ (b) 𝓝 (𝐋) ≜ {𝒙 ∈ 𝙓 |𝐋𝒙 = 𝟘} ⊆ 𝙓 (c) 𝒙 + 𝒚 ∈ 𝓝 (𝐋) ⟹ 𝟘 = 𝐋(𝒙 + 𝒚) = 𝐋(𝒚 + 𝒙) ⟹ 𝒚 + 𝒙 ∈ 𝓝 (𝐋) (d) 𝛼 ∈ 𝔽 , 𝒙 ∈ 𝙓 ⟹ 𝟘 = 𝐋𝒙 ⟹ 𝟘 = 𝛼𝐋𝒙 ⟹ 𝟘 = 𝐋(𝛼𝒙) ⟹ 𝛼𝒙 ∈ 𝓝 (𝐋) 2. Proof that 𝓘(𝐋) is a linear subspace of 𝙔 : (a) 𝟘 ∈ 𝓘(𝐋) ⟹ 𝓘(𝐋) ≠ ∅ (b) 𝓘(𝐋) ≜ {𝒚 ∈ 𝙔 |∃𝒙 ∈ 𝙓

such that

(c) 𝒙 + 𝒚 ∈ 𝓘(𝐋) ⟹ ∃𝒗 ∈ 𝙓

𝒚 = 𝐋𝒙} ⊆ 𝙔

such that

(d) 𝛼 ∈ 𝔽 , 𝒙 ∈ 𝓘(𝐋) ⟹ ∃𝒙 ∈ 𝙓

𝐋𝒗 = 𝒙 + 𝒚 = 𝒚 + 𝒙 ⟹ 𝒚 + 𝒙 ∈ 𝓘(𝐋)

such that

𝒚 = 𝐋𝒙 ⟹ 𝛼𝒚 = 𝛼𝐋𝒙 = 𝐋(𝛼𝒙) ⟹ 𝛼𝒙 ∈ 𝓘(𝐋)

✏ Example D.1. 8 Let 𝒞([𝑎 ∶ 𝑏], ℝ) be the set of all continuous functions from the closed real interval [𝑎 ∶ 𝑏] to ℝ. e x 𝒞([𝑎 ∶ 𝑏], ℝ) is a linear space. 7 8

📘 Michel and Herget (1993) pages 98–104, 📘 Berberian (1961) pages 80–85 ⟨Theorem IV.1.4 and Theorem IV.3.1⟩ 📘 Eidelman et al. (2004) page 3

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Theorem D.3. 9 Let ℒ(𝙓 , 𝙔 ) be the set of linear operators from a linear space 𝙓 to a linear space 𝙔 . Let 𝓝 (𝐋) be the NULL SPACE of a linear operator 𝐋 ∈ ℒ(𝙓 , 𝙔 ). t 𝐋𝒙 = 𝐋𝒚 ⟺ 𝒙 − 𝒚 ∈ 𝓝 (𝐋) h 𝐋 is INJECTIVE ⟺ 𝓝 (𝐋) = {𝟘} m ✎PROOF: 1. Proof that 𝐋𝒙 = 𝐋𝒚 ⟹ 𝒙 − 𝒚 ∈ 𝓝 (𝐋): 𝐋(𝒙 − 𝒚) = 𝐋𝒙 − 𝐋𝒚

by Theorem D.1 page 103

=𝟘 ⟹

by left hypothesis 𝒙 − 𝒚 ∈ 𝓝 (𝐋)

by definition of null space

2. Proof that 𝐋𝒙 = 𝐋𝒚 ⟸ 𝒙 − 𝒚 ∈ 𝓝 (𝐋): 𝐋𝒚 = 𝐋𝒚 + 𝟘

by definition of linear space (Definition D.1 page 101)

= 𝐋𝒚 + 𝐋(𝒙 − 𝒚)

by right hypothesis

= 𝐋𝒚 + (𝐋𝒙 − 𝐋𝒚)

by Theorem D.1 page 103

= (𝐋𝒚 − 𝐋𝒚) + 𝐋𝒙

by associative and commutative properties (Definition D.1 page 101)

= 𝐋𝒙 3. Proof that 𝐋 is injective ⟺ 𝓝 (𝐋) = {𝟘}: 𝐋 is injective ⟺ {(𝐋𝒙 = 𝐋𝒚

⟺

⟺ {[𝐋𝒙 − 𝐋𝒚 = 𝟘 ⟺ {[𝐋(𝒙 − 𝒚) = 𝟘

𝒙 = 𝒚)

∀𝒙, 𝒚 ∈ 𝑋}

⟺

(𝒙 − 𝒚) = 𝟘]

∀𝒙, 𝒚 ∈ 𝑋 }

⟺

(𝒙 − 𝒚) = 𝟘]

∀𝒙, 𝒚 ∈ 𝑋 }

⟺ 𝓝 (𝐋) = {𝟘}

✏ Theorem D.4. 10 Let 𝙒 , 𝙓 , 𝙔 , and 𝙕 be linear spaces over a field 𝔽 . 1. 𝐋(𝐌𝐍) = (𝐋𝐌)𝐍 ∀𝐋∈ℒ(𝙕 ,𝙒 ), 𝐌∈ℒ(𝙔 ,𝙕 ), 𝐍∈ℒ(𝙓 ,𝙔 ) t 2. 𝐋(𝐌 ⨢ 𝐍) = (𝐋𝐌) ⨢ (𝐋𝐍) ∀𝐋∈ℒ(𝙔 ,𝙕 ), 𝐌∈ℒ(𝙓 ,𝙔 ), 𝐍∈ℒ(𝙓 ,𝙔 ) h ∀𝐋∈ℒ(𝙔 ,𝙕 ), 𝐌∈ℒ(𝙔 ,𝙕 ), 𝐍∈ℒ(𝙓 ,𝙔 ) m 3. (𝐋 ⨢ 𝐌)𝐍 = (𝐋𝐍) ⨢ (𝐌𝐍) 4. 𝛼(𝐋𝐌) = (𝛼𝐋)𝐌 = 𝐋(𝛼𝐌) ∀𝐋∈ℒ(𝙔 ,𝙕 ), 𝐌∈ℒ(𝙓 ,𝙔 ), 𝛼∈𝔽

( ASSOCIATIVE) ( LEFT DISTRIBUTIVE) ( RIGHT DISTRIBUTIVE) ( HOMOGENEOUS)

✎PROOF: 1. Proof that 𝐋(𝐌𝐍) = (𝐋𝐌)𝐍: Follows directly from property of associative operators. 2. Proof that 𝐋(𝐌 ⨢ 𝐍) = (𝐋𝐌) ⨢ (𝐋𝐍): [𝐋(𝐌 ⨢ 𝐍)]𝒙 = 𝐋[(𝐌 ⨢ 𝐍)𝒙] = 𝐋[(𝐌𝒙) ⨢ (𝐍𝒙)] = [𝐋(𝐌𝒙)] ⨢ [𝐋(𝐍𝒙)]

by additive property Definition D.5 page 103

= [(𝐋𝐌)𝒙] ⨢ [(𝐋𝐍)𝒙] 9 10

📘 Berberian (1961) page 88 ⟨Theorem IV.1.4⟩ 📘 Berberian (1961) page 88 ⟨Theorem IV.5.1⟩



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3. Proof that (𝐋 ⨢ 𝐌)𝐍 = (𝐋𝐍) ⨢ (𝐌𝐍): Follows directly from property of associative operators. 4. Proof that 𝛼(𝐋𝐌) = (𝛼𝐋)𝐌: Follows directly from associative property of linear operators. 5. Proof that 𝛼(𝐋𝐌) = 𝐋(𝛼𝐌): [𝛼(𝐋𝐌)]𝒙 = 𝛼[(𝐋𝐌)𝒙] = 𝐋[𝛼(𝐌𝒙)]

by homogeneous property Definition D.5 page 103

= 𝐋[(𝛼𝐌)𝒙] = [𝐋(𝛼𝐌)]𝒙

✏

D.2 Operators on Normed linear spaces D.2.1 Operator norm Definition D.6. 11 Let 𝙑 = (𝑋, 𝔽 , ⨣, ⋅) be a linear space and 𝔽 be a field with absolute value function |⋅| ∈ ℝ𝔽 . A norm is any functional ‖⋅‖ in ℝ𝙓 that satisfies 1. ∀𝒙∈𝑋 ( STRICTLY POSITIVE) and ‖𝒙‖ ≥ 0 d 2. ∀𝒙∈𝑋 ( NONDEGENERATE) and ‖𝒙‖ = 0 ⟺ 𝒙 = 𝟘 e 3. = |𝑎| ∀𝒙∈𝑋, 𝑎∈ℂ ( HOMOGENEOUS) and ‖𝑎𝒙‖ ‖𝒙‖ f 4. ‖𝒙 + 𝒚‖ ≤ ‖𝒙‖ + ‖𝒚‖ ∀𝒙,𝒚∈𝑋 ( SUBADDITIVE/triangle inquality). A normed linear space is the pair (𝙑 , ‖⋅‖). Definition D.7. 13

12

Let ℒ(𝙓 , 𝙔 ) be the space of linear operators over normed linear spaces 𝙓 and 𝙔 .

The operator norm ⦀⋅⦀ is defined as ∀𝐀 ∈ ℒ(𝙓 , 𝙔 ) ⦀𝐀⦀ ≜ sup {‖𝐀𝒙‖ | ‖𝒙‖ ≤ 1}

d e f

𝒙∈𝙓

The pair (ℒ(𝙓 , 𝙔 ), ⦀⋅⦀) is the normed space of linear operators on (𝙓 , 𝙔 ).

Proposition D.1 (next) shows that the functional defined in Definition D.7 (previous) is a norm. Proposition D.1. 14 Let (ℒ(𝙓 , 𝙔 ), ⦀⋅⦀) be the normed space of linear operators over the normed linear spaces 𝙓 ≜ (𝑋, +, ⋅, (𝔽 , ∔, ⨰), ‖⋅‖) and 𝙔 ≜ (𝑌 , +, ⋅, (𝔽 , ∔, ⨰), ‖⋅‖). The functional ⦀⋅⦀ is a norm on ℒ(𝙓 , 𝙔 ). In particular, 1. ∀𝐀∈ℒ(𝙓 ,𝙔 ) ( NON-NEGATIVE) and ⦀𝐀⦀ ≥ 0 p 2. ⟺ 𝐀≗𝟘 ∀𝐀∈ℒ(𝙓 ,𝙔 ) ( NONDEGENERATE) and ⦀𝐀⦀ = 0 r ), 3. = |𝛼| ∀𝐀∈ℒ(𝙓 ,𝙔 𝛼∈𝔽 ( HOMOGENEOUS) and ⦀𝛼𝐀⦀ ⦀𝐀⦀ p 4. ⦀𝐀 ⨢ 𝐁⦀ ≤ ⦀𝐀⦀ + ⦀𝐁⦀ ∀𝐀∈ℒ(𝙓 ,𝙔 ) ( SUBADDITIVE). Moreover, (ℒ(𝙓 , 𝙔 ), ⦀⋅⦀) is a normed linear space. 11

📘 Aliprantis and Burkinshaw (1998) pages 217–218, 📘 Banach (1932a) page 53, 📘 Banach (1932b) page 33, 📘 Banach (1922) page 135 12 📘 Rudin (1991) page 92, 📘 Aliprantis and Burkinshaw (1998) page 225 13 The operator norm notation ⦀⋅⦀ is introduced (as a Matrix norm) in 📘 Horn and Johnson (1990), page 290 14 📘 Rudin (1991) page 93 CC

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✎PROOF: 1. Proof that ⦀𝐀⦀ > 0 for 𝐀 ≠ 𝟘: ⦀𝐀⦀ ≜ sup {‖𝐀𝒙‖ | ‖𝒙‖ ≤ 1}

by definition of ⦀⋅⦀ (Definition D.7 page 106)

𝒙∈𝙓

>0 2. Proof that ⦀𝐀⦀ = 0 for 𝐀 ≗ 𝟘: ⦀𝐀⦀ ≜ sup {‖𝐀𝒙‖ | ‖𝒙‖ ≤ 1}


𝒙∈𝙓

= sup {‖𝟘𝒙‖ | ‖𝒙‖ ≤ 1} 𝒙∈𝙓

=0 3. Proof that ⦀𝛼𝐀⦀ = |𝛼| ⦀𝐀⦀: ⦀𝛼𝐀⦀ ≜ sup {‖𝛼𝐀𝒙‖ | ‖𝒙‖ ≤ 1}


𝒙∈𝙓

= sup {|𝛼| ‖𝐀𝒙‖ | ‖𝒙‖ ≤ 1}


𝒙∈𝙓

= |𝛼| sup {‖𝐀𝒙‖ | ‖𝒙‖ ≤ 1}

by definition of sup

𝒙∈𝙓

= |𝛼| ⦀𝐀⦀


4. Proof that ⦀𝐀 ⨢ 𝐁⦀ ≤ ⦀𝐀⦀ + ⦀𝐁⦀: ⦀𝐀 ⨢ 𝐁⦀ ≜ sup {‖(𝐀 ⨢ 𝐁)𝒙‖ | ‖𝒙‖ ≤ 1}


𝒙∈𝙓

= sup {‖𝐀𝒙 + 𝐁𝒙‖ | ‖𝒙‖ ≤ 1} 𝒙∈𝙓

≤ sup {‖𝐀𝒙‖ + ‖𝐁𝒙‖ | ‖𝒙‖ ≤ 1}


𝒙∈𝙓

≤ sup {‖𝐀𝒙‖ | ‖𝒙‖ ≤ 1} + sup {‖𝐁𝒙‖ | ‖𝒙‖ ≤ 1} 𝒙∈𝙓

𝒙∈𝙓

≜ ⦀𝐀⦀ + ⦀𝐁⦀


✏ Lemma D.1. Let (ℒ(𝙓 , 𝙔 ), ⦀⋅⦀) be the normed space of linear operators over normed linear spaces 𝙓 ≜ (𝑋, +, ⋅, (𝔽 , ∔, ⨰), ‖⋅‖) and 𝙔 ≜ (𝑌 , +, ⋅, (𝔽 , ∔, ⨰), ‖⋅‖). l e ⦀𝐋⦀ = sup {‖𝐋𝒙‖ | ‖𝒙‖ = 1} ∀𝒙 ∈ ℒ(𝙓 , 𝙔 ) m 𝒙 ✎PROOF:

15

1. Proof that sup𝒙 {‖𝐋𝒙‖ | ‖𝒙‖ ≤ 1} ≥ sup𝒙 {‖𝐋𝒙‖ | ‖𝒙‖ = 1}: sup {‖𝐋𝒙‖ | ‖𝒙‖ ≤ 1} ≥ sup {‖𝐋𝒙‖ | ‖𝒙‖ = 1} 𝒙 15

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Hi, Dan, OK, let's discuss it in Friday, Btw, there is something I missed in my last mail, the following is more complete than previous one. (a) sup_x { || Lx || : ||x||= sup_x { || Lx || : ||x|| =1 } , since sup of a set is always greater than sup of its subset. (b) Assume\vy_{1},..\vy_{n}.. be a sequence in the set {\vx : ||x||= sup_n { || L z_{n} || } >= sup_n { || L Y_{n} || } = sup_x { || Lx || : ||x||