Square of the Error Octonionic Theorem and Hypercomplex ... - SciELO

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Tendˆencias em Matem´atica Aplicada e Computacional, 14, N. 3 (2013), 483-495 © 2013 Sociedade Brasileira de Matem´atica Aplicada e Computacional www.scielo.br/tema doi: 10.5540/tema.2013.014.03.0483

Square of the Error Octonionic Theorem and Hypercomplex Fourier Serier C.A.P. MARTINEZ1 , A.L.M. MARTINEZ1 , M.F. BORGES2 and E.V. CASTELANI3* Received on March 21, 2013 / Accepted on October 13, 2013

ABSTRACT. The focus of this paper is to address some classical results for a class of hypercomplex numbers. More specifically we present an extension of the Square of the Error Theorem and a Bessel inequality for octonions. Keywords: hypercomplex functions, Fourier series, octonions.

1

INTRODUCTION AND MOTIVATION

Octonions are hypercomplex numbers and in many ways may be regarded as non-associative quaternions. Nowadays they have been used in appropriated approaches of higher dimensional physics [5, 8], such as M-theory, Strings and alternative gravity theories. In this paper, with the main purpose in the nearest future of making a larger application of hypercomplex in unified field theories, and motivated both by earlier works by Eilenberg, Niven [7], Deavours [6], Sinegre [16], and on recent results obtained by some of the authors [2, 3, 4, 12, 13, 14], Boek, Gurlebeck [1], and Lam [11], we have concentrated our efforts in constructing an extension of the Algebra of hypercomplex. Hypercomplex numbers are mainly concerned to quaternions and octonions. Their algebras may regarded as extensions of the ordinary bi-dimensional complex algebra [10, 15], either being non-commutative and associative for quaternions or non-commutative and non-associative in the octonionic case. Quaternions were discovered in 1843 by Willian R. Hamilton (op. cit. [16]), and in that same year John T. Graves, a Hamilton’s friend, found an 8-dimensional Algebra whose property ∗ Corresponding author: Emerson Vitor Castelani 1 Coordenac¸a˜ o de Matem´atica, COMAT, UTFPR – Universidade Tecnol´ogica Federal do Paran´a, Campus Corn´elio Proc´opio, Av. Alberto Carazzai, 1640, 86300-000 Corn´elio Proc´opio, PR, Brasil. E-mails: [email protected]; [email protected] 2 Department of Computing, UNESP – State University of S˜ao Paulo at S˜ao Jos´e do Rio Preto, 15054-000 S˜ao Jos´e do Rio Preto, SP, Brazil. E-mail: [email protected] 3 Departamento de Matem´atica, DMA, UEM – Universidade Estadual de Maring´a, Av. Colombo, 5790, 87020-900 Maring´a, PR, Brasil. E-mail: [email protected]

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on non-associativity in a multiplication table holds. Further on, two years later, in 1845, after some contributions on this subject by Arthur Cayley octonions have also been named “Cayley numbers”. In the context of the present work, and based in a previous result (op. cit. [13]), we extend for octonions the concept of a 2L-periodical function, that we will call Octonionic Fourier Series. We define an octonionic exponential function, and show both a non-associative expansion of the Moivre Theorem and generalizations of the Euler formula. Finally, we obtain octonionic versions of the Square of the Error Theorem and the Bessel inequality. 1.1

Hypercomplex Fourier Series

The octonions are a somewhat nonassociativite extension of the quaternions. They form the 8dimensional normed division algebra on R. The octonionic algebra, also called octaves denoted for O , is an alternative division algebra. The octonions set O is denoted by O = {a, b, c, d, e, f , g, h} ∈ R,

where, (a, b, c, d, e, f, g, h) = (a  , b , c , d  , e , f  , g , h  ) ⇐⇒ a = a  , b = b , c = c , d = d  , e = e , f = f  , g = g , h = h  . The octonions do not form a ring due the non-commutativity of the multiplication. Also do not form a group due a nonassociativide of multiplication. They form a Moufang Loop, a Loop with identity element (Jacobson, N., [9]). Let us consider an octonionic number given by o = a + bi + cj + dk + el + f li + glj + hlk The octonion unities 1, i, j, k, l, li, lj, lk form an orthonormal base of the 8-dimensional algebra. Similarly to [13] we start our contribution in considering f a function defined on interval [−L , L], L > 0, and outside of this interval set as f (x) = f (x + 2L), that is, f (x) is 2L−periodical. If f and f  are piecewise continuous then the series of function given below,  nπ x   nπ x  a0   + an cos + bn sin 2 L L ∞

(1.1)

n=1

is convergent and the limit is lima→x + f (x) + lima→x − f (x)  f (x) = . 2

Tend. Mat. Apl. Comput., 14, N. 3 (2013)

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The coefficients of Fourier of f , a0 , an and bn given by:  1 L a0 = f (x)d x, L −L   nπ x  1 L f (x) cos an = d x, L −L L and bn =

1 L



L −L

f (x) sin

 nπ x  L

d x.

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(1.2)

(1.3)

The trigonometric series presented in (1.1) with this choice of coefficients is the Fourier series of f . Now, we will detail some properties considering octonions o ∈ O given by → o = u 1 + u 2 i + u 3 j + u 4 k + u 5l + u 6li + u 7l j + u 8lk = u 1 + − u. The extended equation of De Moivre for octonions is given by:

 o u1 2 2 2 2 2 2 2 cos e = e u2 + u3 + u4 + u5 + u6 + u7 + u8 

sin( u 2 + u 2 + u 2 + u 2 + u 2 + u 2 + u 2 ) 7 2 3 4 5 6 8 + u  u 22 + u 23 + u 24 + u 25 + u 26 + u 27 + u 28

(1.4)

Then, using (1.4) we can obtain: y −y e0 (cos |y| + i |y| sin |y|) + e0 (cos | − y| + i |−y| sin | − y|) eiy + e−iy = . 2 2

Since cos(−y) = cos(y) and sin(−y) = − sin(y) we have that eiy + e−iy = cos(y). 2 Following the same arguments we can get cos y =

eiy + e−iy e j y + e− j y eky + e−ky ely + e−ly = = = 2 2 2 2

e(l j )y + e−(l j )y e(lk)y + e−(lk)y e(li)y + e−(li)y = = = 2 2 2 and sin y =

(1.5)

e j y − e− j y eky − e−ky ely − e−ly eiy − e−iy = = = 2i 2j 2k 2l

e(li)y − e−(li)y e(l j )y − e−(l j )y e(lk)y − e−(lk)y = = = . 2li 2l j 2lk

(1.6)

Tend. Mat. Apl. Comput., 14, N. 3 (2013)

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From (1.5) and (1.6), we obtain, (eiy + e j y + · · · + e(lk)y ) + (e−iy + e− j y + · · · + e−(lk)y ) , 14   1 eiy − e−iy e j y − e− j y e(lk)y − e−(lk)y . sin y = + +···+ 7 2i 2j 2lk

cos y =

Consequently,

nπ x an cos L  =

an

(ei(

+ bn

=

1 7

nπ x L

)





nπ x + bn sin L nπ x L

+ . . . + e(lk)( 14

)

)



+

(e−i(

nπ x L

)

+ . . . + e−(lk)( 14

nπ x L )

)



 nπ x nπ x nπ x nπ x  1 ei( L ) − e−i( L ) e(lk)( L ) − e−(lk)( L ) + 7 2i 2lk





inπ x −inπ x an bn bn an L + − e + e L + ... 2 2i 2 2i       cn1

1 c−n



(lk)nπ x −(lk)nπ x bn bn an an + − . e L + e L 2 2(lk) 2 2(lk)      

+

cn7

7 c−n

Noting that cn1 =

bn 1 an + = (an − ibn ) 2 2i 2

and using (1.2) and (1.3) we have: cn1 =

1 2L



L −L

f (x)e−i (

nπ x L

) d x.

(1.7)

Again, using the same argument we can get: 1 = c−n

1 2L



L −L

f (x)e

−i



(−n)π x L



d x.

(1.8)

Extending the ideas used, we conclude that: cn2 =

1 2L



L −L

f (x)e− j (

nπ x L

) d x,

(1.9)

Tend. Mat. Apl. Comput., 14, N. 3 (2013)

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and 2 c−n =

since ( j )2 = −1. cn3 =



1 2L

1 2L



1 2L

cn4 =



cn5 =

1 2L



cn6

1 = 2L

cn7

1 = 2L

f (x)ek(

nπ x L

) d x,

(1.12)

f (x)e−l (

nπ x L

) d x,

(1.13)

f (x)el (

nπ x L

) d x,

(1.14)

f (x)e−li (

nπ x L

) d x,

(1.15)

f (x)eli (

nπ x L

) d x,

(1.16)

f (x)e−l j (

nπ x L

) d x,

(1.17)

f (x)el j (

nπ x L

) d x,

(1.18)

f (x)e−lk(

nπ x L

) d x,

(1.19)

f (x)elk(

nπ x L

) d x,

(1.20)

L

L

L



L −L



L −L



and 7 c−n

(1.11)

−L

1 2L

since (l j )2 = −1.

) d x,

−L



and 6 = c−n

nπ x L

L



1 2L

since (li)2 = −1.

f (x)e−k(

−L

and 5 = c−n

(1.10)

−L

1 = 2L

since (l)2 = −1.

) d x,

−L

and 4 c−n

nπ x L

L

1 = 2L

since (k)2 = −1.

f (x)e j (

−L

and 3 c−n

L

1 = 2L



L −L

L −L



487

L −L

since (lk)2 = −1. From (1.7), (1.8), (1.9), (1.10), (1.11), (1.12), (1.13), (1.14),(1.15), (1.16), (1.17), (1.18), (1.19) and (1.20) we can get 



∞   nπ x nπ x + bn sin an cos L L n=1

=

1 7

 ∞ 

cn1 e

inπ x L

+ cn2 e

j nπ x L

+ cn3 e

knπ x L

+ cn4 e

lnπ x L

+ cn5 e

linπ x L

+ cn6 e

l j nπ x L

+ cn7 e

lknπ x L



−∞,n =0

Tend. Mat. Apl. Comput., 14, N. 3 (2013)

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Since f : R → R is periodic, with period 2L, f and f  are piecewise continuous, we have that the Fourier series of f presented in (1.1) can be written as follows 1  f (x) = co + 7 + cn4 e where considering c0 =

 ∞

cn1 e

inπ x L

+ cn2 e

j nπ x L

+ cn3 e

knπ x L

−∞,n =0 lnπ x L

+ cn5 e

linπ x L

+ cn6 e

l j nπ x L

+ cn7 e

lknπ x L

(1.21)

,

a0 2.

The series (1.21) is the octonionic Fourier series of f . Example 1. Let f (x) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

2,

if 0 ≤ x < π

0,

if − π < x ≤ 0 .

f (x + 2π) = 0.

We will determine the octonionic Fourier series of f . In fact, calculating the coefficients c0 , cn1 , cn2 , cn3 , cn4 , cn5 , cn7 , cn7 we have    −L  π 0 1 a0 1 c0 = = f (x)d x = 0d x + 2d x = 1, 2 2L L 2π −π 0 cn1

=

cn2

=

cn3

=

cn4

=

cn5

=

cn6

=

cn7

=

1 2π 1 2π 1 2π 1 2π 1 2π 1 2π 1 2π



π

2e−inx d x =

i i [cos(nπ) − 1] = [(−1)n − 1], nπ nπ

2e− j nx d x =

j j [cos(nπ) − 1] = [(−1)n − 1], nπ nπ

2e−knx d x =

k k [cos(nπ) − 1] = [(−1)n − 1], nπ nπ

2e−lnx d x =

l l [cos(nπ) − 1] = [(−1)n − 1], nπ nπ

0



π 0



π 0



π 0



π

2e−(li)nx d x =

(li) (li) [cos(nπ) − 1] = [(−1)n − 1], nπ nπ

2e−(l j )nx d x =

(l j ) (l j ) [cos(nπ) − 1] = [(−1)n − 1], nπ nπ

2e−(lk)nx d x =

(lk) (lk) [cos(nπ) − 1] = [(−1)n − 1], nπ nπ

0



π 0



π 0

Tend. Mat. Apl. Comput., 14, N. 3 (2013)

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thus  f (x) = 1 +

 ∞  j [(−1)n − 1] j nx i[(−1)n − 1] inx e + e nπ nπ

−∞,n =0

k[(−1)n − 1] knx l[(−1)n − 1] lnx (li)[(−1)n − 1] (li)nx e e + e + nπ nπ nπ  (l j )[(−1)n − 1] (l j )nx (lk)[(−1)n − 1] (lk)nx . + + e e nπ nπ +

Example 2. Let f (t ) = t , with t ∈ (−1, 1) and f (t + 2) = f (t ). The octonionic coefficients are given by: c0

=

cn1

=

cn2 cn3

= =

0,  1 1 −inπt (−1)n+1 , te dt = 2 −1 (inπ) 1 2 1 2

cn4

=

1 2

cn5

=

1 2

cn6

=

1 2

cn7

=

1 2



1 −1



1 −1



1 −1



1 −1



1 −1



1 −1

t e− j nπt dt =

(−1)n+1 , ( j nπ)

t e−knπt dt =

(−1)n+1 , (knπ)

t e−lnπt dt =

(−1)n+1 , (lnπ)

t e−(li)nπt dt =

(−1)n+1 , ((li)nπ)

t e−(l j )nπt dt =

(−1)n+1 , ((l j )nπ)

t e−(lk)nπt dt =

(−1)n+1 , ((lk)nπ)

Then, the octonionic Fourier series of f is  ∞  (−1)n+1 inπt (−1)n+1 j nπt  e e + f (t ) = inπ j nπ −∞,n =0

(−1)n+1 knπt (−1)n+1 lnπt (−1)n+1 (li)nπt e e e + + knπ lnπ (li)nπ  (−1)n+1 (l j )nπt (−1)n+1 (lk)nπt e e + + (l j )nπ (lk)nπ +

Tend. Mat. Apl. Comput., 14, N. 3 (2013)

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AN EXTENSION OF THE SQUARE OF THE ERROR THEOREM FOR OCTONIONIC FOURIER SERIES AND THE BESSEL INEQUALITY

In this Section we will present an important property of the Hypercomplex Fourier Series derived from some results from the classical Complex Analysis [10]. We also make an extension of the so called Square of the Error Theorem (Walter Rudin [15]). For the purposes of this section, we denote φ the octonionic function defined by φ : R → O, φ(x) = φ1(x) + φ2 (x)i + φ3(x) j + φ4 (x)k + φ5 (x)l + φ6 (x)li + φ7 (x)l j + φ8 (x)lk where φ1 , φ2 , φ3 , φ4 , φ5, φ6 , φ7 and φ8 are real functions. Thus we can consider φ(x) = φ1 (x) − [φ2 (x)i + φ3 (x) j + φ4(x)k + φ5 (x)l + φ6 (x)li + φ7 (x)l j + φ8(x)lk], |φ(x)| = φ(x)φ(x) = 2

8 

(φm (x))2

m=1

Definition 2.1. Let {φn }, n = 1, 2, 3, 4, . . . be a sequence of octonionic functions on [a, b], such that  b φn (x)φm (x)d x = 0, (n = m). (2.1) a

Then, we say that {φn } is an orthogonal system of octonionic functions on [a, b]. In addition, if  b |φn (x)|2 d x = 1, (2.2) a

for all n, we say that {φn } is orthonormal. Here are some examples. Example 3. The sequence of functions φn1 (x)

=

1 inπ x √ e L , n = 1, 2, 3, . . . 2L

φn2 (x)

=

j nπ x 1 √ e L , n = 1, 2, 3, . . . 2L

φn3 (x)

=

1 knπ x √ e L , n = 1, 2, 3, . . . 2L

φn4 (x)

=

1 lnπ x √ e L , n = 1, 2, 3, . . . 2L

φn5 (x)

=

1 linπ x √ e L , n = 1, 2, 3, . . . 2L Tend. Mat. Apl. Comput., 14, N. 3 (2013)

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φn6 (x)

=

1 l j nπ x √ e L , n = 1, 2, 3, . . . 2L

φn7 (x)

=

1 lknπ x √ e L , n = 1, 2, 3, . . . 2L

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form orthonormal system on [−L , L]. Example 4. The sequence of functions

j nπ x l j nπ x inπ x knπ x lnπ x linπ x lknπ x 1 φn (x) = √ e L +e L +e L +e L +e L +e L +e L , 56L for n = 1, 2, 3, . . . form orthonormal system on [−L , L]. Motivated by example 4, we define cn1 + cn2 i + cn3 j + cn4 k + cn5 l + cn6li + cn7 l j + cn8 lk  b f (t )φn (t )dt, for all n = 1, 2, 3, . . . , =

(2.3)

a

where {φn } is an orthonormal sequence in [a, b] and cn1 , cn2 , cn3 , cn4 , cn5 , cn6 , cn7 and cn8 are real sequences. We call cn the n-th octonionic Fourier coefficient of f (relative to {φn }). We write ∞ 

(cn1 + cn2 i + cn3 j + cn4 k + cn5 l + cn6 li + cn7l j + cn8 lk)φn (x)

(2.4)

1

and call this series the octonionic Fourier series of f . The following theorem extends some classical results (op. cit. [15]), which have already been extended for the quaternionic case (op. cit. [13]). More specifically we show that partial sums of octonionic Fourier Series of a function f , have certain minimum property. Let us assume that f is a real function. Theorem 2.1. Let {φn } be orthonormal sequence of octonionic functions on [a, b]. Consider the n-th partial sum of the octonionic Fourier series of f sn (x) =

n  

 1 2 3 4 5 6 7 8 + cm i + cm j + cm k + cm l + cm li + cm l j + cm lk φm (x), cm

(2.5)

m=1 1 , c2 , c3 , c4 , c5 , c6 , c7 and c8 are given in (2.3). In addition, define where cm m m m m m m m

tn (x) =

n  

 dm1 + dm2 i + dm3 j + dm4 k + dm5 l + dm6 li + dm7 l j + dm8 lk φm (x),

(2.6)

m=1

Tend. Mat. Apl. Comput., 14, N. 3 (2013)

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where dm1 , dm2 , dm3 , dm4 , dm5 , dm6 , dm7 and dm8 are real sequences. Then 

b

 | f (x) − sn (x)| d x ≤ 2

a

b

| f (x) − tn (x)|2 d x,

(2.7)

a

and equality holds if and only if 1 2 3 4 5 dm1 = cm , dm2 = cm , dm3 = cm , dm4 = cm , dm5 = cm ,

(2.8)

6 7 8 dm6 = cm , dm7 = cm and dm8 = cm (m = 1, 2, . . . , n).

That means, among all functions tn , sn gives the best possible mean square approximation to f .   Proof. To simplify the notation, let be the integral over [a, b] and the sum from 1 to n. From the definition of (2.6) and (2.3), we have     1 2 3 4 5 6 7 8 (dm + dm i + dm j + dm k + dm l + dm li + dm l j + dm lk)φm f tn = f = =

 

Consequently, 

 (dm1 + dm2 i + dm3 j + dm4 k + dm5 l + dm6 li + dm7 l j + dm8 lk)

f φm

1 2 7 8 (dm1 + dm2 i + . . . + dm7 l j + dm8 lk)(cm + cm i + . . . + cm l j + cm lk).

 f tn +

f tn =

 1 2 3 4 ) + 2(dm2 cm ) + 2(dm3 cm ) + 2(dm4 cm ) 2(dm1 cm  5 6 7 8 + 2(dm5 cm ) + 2(dm6 cm ) + 2(dm7 cm ) + 2(dm8 cm ).

Furthermore,  |tn |2 = =

 tn tn   n n  1    (dm + . . . + dm8 lk)φm (d 1p + . . . + d 8p lk)φ p m=1

=

p=1

n   1 2  (dm ) + (dm2 )2 + (dm3 )2 + (dm4 )2 + (dm5 )2 + (dm6 )2 + (dm7 )2 + (dm8 )2 m=1

since {φn } is orthonormal. Therefore,      2 2 |f| − f tn − f tn + |tn |2 | f − tn | =  =

| f |2 −

  1 2 7 8 ) + 2(dm2 cm ) + . . . + 2(dm7 cm ) + 2(dm8 cm ) 2(dm1 cm

Tend. Mat. Apl. Comput., 14, N. 3 (2013)

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+ 

493

n   1 2  (dm ) + (dm2 )2 + . . . + (dm7 )2 + (dm8 )2 m=1

| f |2 +

=

i

 1 2 2 2 7 2 ) + (dm2 − cm ) + . . . + (dm7 − cm ) (dm1 − cm

n    1 2  8 2 2 2 7 2 8 2 + (dm8 − cm ) − ) + . . . + (cm ) + (cm ) (cm ) + (cm m=1

which is minimized if and only if 1 2 3 4 5 dm1 = cm , dm2 = cm , dm3 = cm , dm4 = cm , dm5 = cm , 6 7 8 dm6 = cm , dm7 = cm and dm8 = cm (m = 1, 2, . . . , n).



Corollary 2.1. Let {φn } be orthonormal sequence of octonionic functions on [a, b], and if f (x) ∼

∞ 

1 2 3 4 5 6 7 8 (cm + cm i + cm j + cm k + cm l + cm li + cm l j + cm lk)φm (x)

m=1

then

∞ 

 1 2 2 2 7 2 8 2 [(cm ) + (cm ) + . . . + (cm ) + (cm ) ]≤

m=1

b

| f (x)|2 d x.

(2.9)

a

Proof. In the proof of Theorem (2.1) we found     1 2 2 2 7 2 8 2 | f |2 + ) + (dm2 − cm ) + (dm7 − cm ) + (dm8 − cm ) | f − t n |2 = (dm1 − cm −

n  

1 2 2 2 7 2 8 2 ) + (cm ) + . . . + (cm ) + (cm ) (cm



m=1

putting dm1 8 in dm8 = cm

1, cm

2 , d 3 = c3 , d 4 = c4 , d 5 = c5 , d 6 = c6 , d 7 = c7 and = = cm m m m m m m m m m m this calculation, we obtain  b n   1 2  2 2 7 2 8 2 (cm ) + (cm ) + . . . + (cm ) + (cm ) ≤ | f (x)|2 d x, (2.10)

dm2

m=1

a

 since | f −tn |2 ≥ 0. Letting n → ∞ in (2.10), we obtain (2.9), this inequality is a generalization for octonions of Bessel.  3

CONCLUDING REMARKS

In this paper we have discussed on an extended version of Fourier Series for octonions. Fourier Series are remarkable in the way through an orthonormal basis they approximate functions associated to physical problems such as conducting heat and vibrations. Tend. Mat. Apl. Comput., 14, N. 3 (2013)

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SQUARE OF THE ERROR OCTONIONIC THEOREM AND HYPERCOMPLEX FOURIER SERIER

Furthermore, as not few models of Theoretical Physics may be analysed through the geometry and algebra of hypercomplex, it will be our concern to concentrate the next steps in making all possible applications of our results in the context of unified physical theories for higher dimensional space-times. RESUMO. O foco deste trabalho e´ abordar alguns resultados cl´assicos para uma certa classe de n´umeros hipercomplexos. Mais espec´ıficamente, apresentamos uma extens˜ao do Teorema do Erro Quadr´atico e da Desigualdade de Bessel para octˆonios. Palavras-chave: Func¸o˜ es hipercomplexas, S´eries de Fourier, octˆonios.

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