Research Article On a Pointwise Convergence of

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Apr 3, 2014 - which leads to quasi-periodic-rational (QPR) interpolation. We investigate the pointwise convergence of the QPR interpolation in the intervalΒ ...
Hindawi Publishing Corporation International Journal of Analysis Volume 2014, Article ID 249513, 10 pages http://dx.doi.org/10.1155/2014/249513

Research Article On a Pointwise Convergence of Quasi-Periodic-Rational Trigonometric Interpolation Arnak Poghosyan and Lusine Poghosyan Institute of Mathematics of National Academy of Sciences of Armenia, Bagramian Avenue 24/5, 0019 Yerevan, Armenia Correspondence should be addressed to Arnak Poghosyan; [email protected] Received 26 November 2013; Accepted 7 March 2014; Published 3 April 2014 Academic Editor: Nicolas Crouseilles Copyright Β© 2014 A. Poghosyan and L. Poghosyan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce a procedure for convergence acceleration of the quasi-periodic trigonometric interpolation by application of rational corrections which leads to quasi-periodic-rational trigonometric interpolation. Rational corrections contain unknown parameters whose determination is important for realization of interpolation. We investigate the pointwise convergence of the resultant interpolation for special choice of the unknown parameters and derive the exact constants of the main terms of asymptotic errors.

1. Introduction

Let

The quasi-periodic (QP) interpolation 𝐼𝑁,π‘š (𝑓, π‘₯), π‘š β‰₯ 0 (π‘š is integer) and π‘₯ ∈ [βˆ’1, 1], interpolates function 𝑓 on equidistant grid π‘˜ π‘₯π‘˜ = , 𝑁

|𝑛| ≀ 𝑁,

π‘˜ = 0, . . . , π‘ž,

∞

|π‘˜| ≀ 𝑁

(1)

and is exact for a quasi-periodic function π‘’π‘–πœ‹π‘›πœŽπ‘₯ ,

𝐴 π‘ π‘˜ (𝑓) = 𝑓(π‘˜) (1) βˆ’ (βˆ’1)π‘˜+𝑠 𝑓(π‘˜) (βˆ’1) ,

2𝑁 𝜎= 2𝑁 + π‘š + 1

(2)

with period 2/𝜎 which tends to 2 as 𝑁 β†’ ∞. The idea of the QP interpolation is introduced in [1, 2] where it is investigated based on the results of numerical experiments. Explicit representation of the interpolation is derived in [3–5]. There, the convergence of the interpolation is considered in the framework of the 𝐿 2 (βˆ’1, 1)-norm and at the endpoints π‘₯ = Β±1 in terms of the limit function. Pointwise convergence in the interval (βˆ’1, 1) is explored in [6]. The main results there, which we need for further comparison, are the following theorems.

(βˆ’1)π‘Ÿ(π‘š+1) . π‘˜+1 π‘Ÿ=βˆ’βˆž (2π‘Ÿ + π‘₯)

Ξ¦π‘˜,π‘š (π‘’π‘–πœ‹π‘₯ ) = 𝑒(π‘–πœ‹/2)(π‘šβˆ’1)π‘₯ βˆ‘

(3)

We denote by 𝑅𝑁,π‘š (𝑓, π‘₯) the error of the QP interpolation as follows:

𝑅𝑁,π‘š (𝑓, π‘₯) = 𝑓 (π‘₯) βˆ’ 𝐼𝑁,π‘š (𝑓, π‘₯) .

(4)

Theorem 1 (see [6]). Let 𝑓(π‘ž+2π‘š) ∈ 𝐴𝐢[βˆ’1, 1] for some π‘š β‰₯ 1, π‘ž β‰₯ 0, and

𝑓(π‘˜) (βˆ’1) = 𝑓(π‘˜) (1) = 0,

π‘˜ = 0, . . . , π‘ž βˆ’ 1.

(5)

2

International Journal of Analysis

Then, the following estimate holds for |π‘₯| < 1 as 𝑁 β†’ ∞:

where π‘š

𝑅𝑁,π‘š (𝑓, π‘₯) = πΆπ‘ž,π‘š (𝑓)

̌ βˆ’ βˆ‘πœƒ π‘“ΜŒ 𝐹𝑛,π‘š = 𝑓𝑛,π‘š 𝑛,β„“ β„“+𝑁,π‘š ,

𝑁

(βˆ’1) π‘π‘ž+π‘š+1

β„“=1

̌ = 𝑓𝑛,π‘š

[π‘š/2]

π‘šβˆ’π‘˜ Γ— [sin (πœ‹ (𝑁 + 1) 𝜎π‘₯) βˆ‘ ( ) π‘˜ π‘˜=0

Γ—

(11)

π‘š

βˆ’1 2π‘–πœ‹π‘›(π‘ βˆ’1)/(2𝑁+π‘š+1) πœƒπ‘›,β„“ = 𝑒2π‘–πœ‹(β„“+π‘βˆ’π‘›)(𝑁+π‘š)/(2𝑁+π‘š+1) βˆ‘Vβ„“,𝑠 𝑒 .

π‘˜

(βˆ’1) βˆ’ sin (πœ‹π‘πœŽπ‘₯) 22π‘˜+1 cos2π‘˜+2 (πœ‹π‘₯/2)

𝑠=1

βˆ’1 Here, Vβ„“,𝑠

are the elements of the inverse of the Vandermonde matrix (V𝑠,β„“ ) as

[π‘š/2]βˆ’1

π‘šβˆ’π‘˜βˆ’2 Γ— βˆ‘ ( ) π‘˜ π‘˜=0

V𝑠,β„“ = π›Όβ„“π‘ βˆ’1 ,

π‘˜

(βˆ’1) ] + π‘œ (π‘βˆ’π‘žβˆ’π‘šβˆ’1 ) , 22π‘˜+3 cos2π‘˜+4 (πœ‹π‘₯/2) (6)

Γ—

𝑁 1 π‘˜ βˆ‘ 𝑓 ( ) π‘’βˆ’2π‘–πœ‹π‘›π‘˜/(2𝑁+π‘š+1) , 2𝑁 + π‘š + 1 π‘˜=βˆ’π‘ 𝑁

where

𝛼ℓ = 𝑒2π‘–πœ‹(β„“+𝑁)/(2𝑁+π‘š+1) ,

(12)

and have the following explicit form [8]: βˆ’1 =βˆ’ Vβ„“,𝑠

1

𝛼ℓ𝑠 βˆπ‘š 𝑖=1, 𝑖 =ΜΈ β„“

π‘ βˆ’1

𝑗

βˆ‘π›½ 𝛼 , (𝛼ℓ βˆ’ 𝛼𝑖 ) 𝑗=0 𝑗 β„“

β„“, 𝑠 = 1, . . . , π‘š, (13)

where 𝛽𝑗 are the coefficients of the following polynomial: π‘ž

πΆπ‘ž,π‘š (𝑓) = βˆ‘

𝐴 π‘˜π‘ž (𝑓) (π‘š + 1)π‘žβˆ’π‘˜

π‘žβˆ’π‘˜+1 π‘–π‘˜βˆ’1 πœ‹π‘˜βˆ’π‘š+1 π‘˜=0 2

Theorem 2 (see [6]). Let 𝑓 and

(π‘ž+1)

(π‘ž βˆ’ π‘˜)!

Ξ¦(π‘š) π‘˜,π‘š

(βˆ’1) .

(7)

∈ 𝐴𝐢[βˆ’1, 1] for some π‘ž β‰₯ 0

𝑓(π‘˜) (βˆ’1) = 𝑓(π‘˜) (1) = 0,

π‘˜ = 0, . . . , π‘ž βˆ’ 1.

(8)

Then, the following estimate holds for |π‘₯| < 1 as 𝑁 β†’ ∞: 𝑅𝑁,0 (𝑓, π‘₯) = 𝐴 0π‘ž (𝑓) [π‘ž/2]

Γ— βˆ‘ π‘˜=0

(9)

In the current paper, we consider convergence acceleration of the QP interpolation by rational corrections in terms of π‘’π‘–πœ‹πœŽπ‘₯ which leads to quasi-periodic-rational (QPR) interpolation. We investigate the pointwise convergence of the QPR interpolation in the interval (βˆ’1, 1) and derive the exact constants of the main terms of asymptotic errors. Comparison with Theorems 1 and 2 shows the accelerated convergence for smooth functions. Some results of this research are reported also in [7]. More specifically, the QP interpolation can be realized by the following formula: 𝑁

𝑛=βˆ’π‘

𝑗=1

𝑗=0

get

(14)

Taking into account that 𝛼𝑠 βˆ’ 𝛼𝑖 = 𝑂(1/𝑁), from (13), we βˆ’1 Vβ„“,𝑠 = 𝑂 (π‘π‘šβˆ’1 ) ,

𝑁 󳨀→ ∞,

(15)

πœƒπ‘›,β„“ = 𝑂 (π‘π‘šβˆ’1 ) ,

𝑁 󳨀→ ∞.

(16)

In this section, we consider convergence acceleration of the QP interpolation by rational trigonometric corrections which leads to the QPR interpolation. 𝑝 Consider a vector πœ‡ = {πœ‡π‘˜ }π‘˜=1 . By Ξ”π‘˜π‘› (πœ‡, 𝑐𝑛 ), we denote generalized finite differences defined by the following recurrent relations: Ξ”0𝑛 (πœ‡, 𝑐𝑛 ) = 𝑐𝑛 ,

+ π‘œ (π‘βˆ’π‘žβˆ’1 ) .

𝐼𝑁,π‘š (𝑓, π‘₯) = βˆ‘ 𝐹𝑛,π‘š π‘’π‘–πœ‹π‘›πœŽπ‘₯ ,

π‘š

2. Quasi-Periodic-Rational Interpolation

(βˆ’1)𝑁 sin (πœ‹π‘π‘₯) 2π‘ž+1 π‘π‘ž+1 cos (πœ‹π‘₯/2)

∞ (βˆ’1)π‘˜ 22π‘˜ (βˆ’1)𝑠 βˆ‘ (π‘ž βˆ’ 2π‘˜)!πœ‹2π‘˜+1 𝑠=βˆ’βˆž (2𝑠 + 1)2π‘˜+1

π‘š

∏ (π‘₯ βˆ’ 𝛼𝑗 ) = βˆ‘ 𝛽𝑗 π‘₯𝑗 .

π‘š ∈ 𝑍, π‘š β‰₯ 0,

(10)

π‘˜βˆ’1 Ξ”π‘˜π‘› (πœ‡, 𝑐𝑛 ) = Ξ”π‘˜βˆ’1 𝑛 (πœ‡, 𝑐𝑛 ) + πœ‡π‘˜ Ξ” π‘›βˆ’1 (πœ‡, 𝑐𝑛 ) ,

π‘˜ β‰₯ 1,

(17)

for some sequence 𝑐𝑛 . When πœ‡ ≑ 1, we put Ξ”π‘˜π‘› (πœ‡, 𝑐𝑛 ) = Ξ”π‘˜π‘› (𝑐𝑛 ) .

(18)

It is easy to verify that Ξ”1𝑛 (πœ‡, 𝑐𝑛 ) = 𝑐𝑛 + πœ‡1 π‘π‘›βˆ’1 , Ξ”2𝑛 (πœ‡, 𝑐𝑛 ) = Ξ”1𝑛 (πœ‡, 𝑐𝑛 ) + πœ‡2 Ξ”1π‘›βˆ’1 (πœ‡, 𝑐𝑛 ) = 𝑐𝑛 + (πœ‡1 + πœ‡2 ) π‘π‘›βˆ’1 + πœ‡1 πœ‡2 π‘π‘›βˆ’2 , Ξ”3𝑛 (πœ‡, 𝑐𝑛 ) = 𝑐𝑛 + (πœ‡1 + πœ‡2 + πœ‡3 ) π‘π‘›βˆ’1 + (πœ‡1 πœ‡2 + πœ‡1 πœ‡3 + πœ‡2 πœ‡3 ) π‘π‘›βˆ’2 + πœ‡1 πœ‡2 πœ‡3 π‘π‘›βˆ’3 . (19)

International Journal of Analysis

3

In general, we can prove by the mathematical induction that

where

𝑝

Δ𝑝𝑛 (πœ‡, 𝑐𝑛 ) = βˆ‘π›Ύπ‘  (πœ‡) π‘π‘›βˆ’π‘  ,

𝛼

𝑗 𝑓(𝑗) (βˆ’1) π‘₯ ( + 1) , 𝑗! 𝜎 𝑗=0

(20)

𝑓left (π‘₯) = βˆ‘

𝑠=0

where 𝛾𝑠 (πœ‡) are the coefficients of the following polynomial: 𝑝

𝑝

𝑠=0

𝑠=1

βˆ‘π›Ύπ‘  (πœ‡) π‘₯𝑠 = ∏ (1 + πœ‡π‘  π‘₯) .

𝛼

𝑗 𝑓(𝑗) (1) π‘₯ 𝑓right (π‘₯) = βˆ‘ ( βˆ’ 1) . 𝑗! 𝜎 𝑗=0

(21) 𝑝 {πœ† π‘˜ }|π‘˜|=1 ,

+

According to definition of π‘“βˆ— , we can write

Consider the following vectors: πœ† = πœ† = 𝑝 𝑝 {πœ† π‘˜ }π‘˜=1 , and πœ†βˆ’ = {πœ† βˆ’π‘˜ }π‘˜=1 . By π›Ώπ‘›π‘˜ (πœ†, 𝑐𝑛 ), we denote modified finite differences defined by the following recurrent relations: 𝛿𝑛0 (πœ†, 𝑐𝑛 ) = 𝑐𝑛 , π›Ώπ‘›π‘˜

(πœ†, 𝑐𝑛 ) =

π›Ώπ‘›π‘˜βˆ’1

(πœ†, 𝑐𝑛 ) +

π‘˜βˆ’1 πœ† βˆ’π‘˜ π›Ώπ‘›βˆ’1

∞

π‘“βˆ— (π‘₯) = βˆ‘ π‘“π‘›βˆ— π‘’π‘–πœ‹π‘›π‘₯ , 𝑛=βˆ’βˆž

∞

𝑓 (π‘₯) = βˆ‘ π‘“π‘›βˆ— π‘’π‘–πœ‹π‘›πœŽπ‘₯ ,

π‘˜ β‰₯ 1, (22)

𝑛=βˆ’βˆž

(23)

𝑝

𝑝

(24)

𝑠=0

where

𝑁

𝑝

𝑝

π‘˜=1

π‘˜=0

𝑝

𝑝

π‘˜=1

π‘˜=0

|𝑛|>𝑁

The following transformation is easy to verify (see details in [9] for similar transformation): 𝑅𝑁,π‘š (𝑓, π‘₯) = πœ† βˆ’1

∏ (1 + πœ† π‘˜ π‘₯) = βˆ‘ π›Ύπ‘˜ (πœ†+ ) π‘₯π‘˜ ,

πΉβˆ’π‘βˆ’1,π‘š π‘’βˆ’π‘–πœ‹πœŽπ‘π‘₯ βˆ’ 𝐹𝑁,π‘š π‘’π‘–πœ‹πœŽ(𝑁+1)π‘₯ (1 + πœ† βˆ’1 π‘’π‘–πœ‹πœŽπ‘₯ ) (1 + πœ† 1 π‘’βˆ’π‘–πœ‹πœŽπ‘₯ )

+ πœ†1

(25)

∏ (1 + πœ† βˆ’π‘˜ π‘₯) = βˆ‘ π›Ύπ‘˜ (πœ†βˆ’ ) π‘₯π‘˜ . +

It is easy to verify that

𝐹𝑁+1,π‘š π‘’π‘–πœ‹πœŽπ‘π‘₯ βˆ’ πΉβˆ’π‘,π‘š π‘’βˆ’π‘–πœ‹πœŽ(𝑁+1)π‘₯ (1 + πœ† βˆ’1 π‘’π‘–πœ‹πœŽπ‘₯ ) (1 + πœ† 1 π‘’βˆ’π‘–πœ‹πœŽπ‘₯ ) 1

(1 + πœ† βˆ’1

π‘’π‘–πœ‹πœŽπ‘₯ ) (1

+ πœ† 1 π‘’βˆ’π‘–πœ‹πœŽπ‘₯ )

(34)

𝑁

π›Ώπ‘›π‘˜ (𝑐𝑛 ) = Ξ”2π‘˜ 𝑛+π‘˜ (𝑐𝑛 ) ,

(26)

Γ— βˆ‘ 𝛿𝑛1 (πœ†, π‘“π‘›βˆ— βˆ’ 𝐹𝑛,π‘š ) π‘’π‘–πœ‹π‘›πœŽπ‘₯

(27)

+

𝑛=βˆ’π‘

π‘˜

π‘˜ Ξ”π‘˜π‘› (𝑐𝑛 ) = βˆ‘ ( ) π‘π‘›βˆ’π‘  , 𝑠 𝑠=0 2π‘˜ (𝑐𝑛 ) = βˆ‘ ( ) 𝑐𝑛+π‘˜βˆ’π‘  . 𝑠 𝑠=0

𝑓left (π‘₯) , π‘₯ ∈ [βˆ’1, βˆ’πœŽ) , { { { π‘₯ π‘₯ ∈ [βˆ’πœŽ, 𝜎] , 𝑓 (π‘₯) = {𝑓 ( ) , { { 𝜎 {𝑓right (π‘₯) , π‘₯ ∈ (𝜎, 1] ,

1 (1 + πœ† βˆ’1 π‘’π‘–πœ‹πœŽπ‘₯ ) (1 + πœ† 1 π‘’βˆ’π‘–πœ‹πœŽπ‘₯ )

Γ— βˆ‘ 𝛿𝑛1 (πœ†, π‘“π‘›βˆ— ) π‘’π‘–πœ‹π‘›πœŽπ‘₯ .

2π‘˜

(28)

We assume that 𝑓 ∈ 𝐢𝛼 [βˆ’1, 1] for some 𝛼 β‰₯ 1 and we denote βˆ—

(32)

𝑛=βˆ’π‘

𝛿𝑛𝑝 (πœ†, 𝑐𝑛 ) = βˆ‘ π›Ύπ‘˜ (πœ†+ ) βˆ‘π›Ύπ‘  (πœ†βˆ’ ) 𝑐𝑛+π‘˜βˆ’π‘  ,

π›Ώπ‘›π‘˜

π‘₯ ∈ [βˆ’1, 1] .

𝑅𝑁,π‘š (𝑓, π‘₯) = βˆ‘ (π‘“π‘›βˆ— βˆ’ 𝐹𝑛,π‘š ) π‘’π‘–πœ‹π‘›πœŽπ‘₯ + βˆ‘ π‘“π‘›βˆ— π‘’π‘–πœ‹π‘›πœŽπ‘₯ . (33)

Similar to (20), we can show that

π‘˜=0

(31)

Therefore,

for some sequence 𝑐𝑛 . When πœ† ≑ 1, we put π›Ώπ‘›π‘˜ (πœ†, 𝑐𝑛 ) = π›Ώπ‘›π‘˜ (𝑐𝑛 ) .

π‘₯ ∈ (βˆ’1, 1) .

Hence,

(πœ†, 𝑐𝑛 )

π‘˜βˆ’1 (πœ†, 𝑐𝑛 ) + πœ† βˆ’π‘˜ π›Ώπ‘›π‘˜βˆ’1 (πœ†, 𝑐𝑛 )) , + πœ† π‘˜ (𝛿𝑛+1

(30)

|𝑛|>𝑁

Reiteration of it up to 𝑝 times leads to the following expansion of the error: 𝑅𝑁,π‘š (𝑓, π‘₯)

(29)

𝑝

= βˆ‘ πœ† βˆ’π‘˜ π‘˜=1

π‘˜βˆ’1 π‘˜βˆ’1 π›Ώβˆ’π‘βˆ’1 (πœ†, 𝐹𝑛,π‘š ) π‘’βˆ’π‘–πœ‹πœŽπ‘π‘₯ βˆ’ 𝛿𝑁 (πœ†, 𝐹𝑛,π‘š ) π‘’π‘–πœ‹πœŽ(𝑁+1)π‘₯

βˆπ‘˜π‘ =1 (1 + πœ† βˆ’π‘  π‘’π‘–πœ‹πœŽπ‘₯ ) (1 + πœ† 𝑠 π‘’βˆ’π‘–πœ‹πœŽπ‘₯ )

4

International Journal of Analysis 𝑝

+ βˆ‘πœ†π‘˜

π‘˜βˆ’1 π‘˜βˆ’1 𝛿𝑁+1 (πœ†, 𝐹𝑛,π‘š ) π‘’π‘–πœ‹πœŽπ‘π‘₯ βˆ’ π›Ώβˆ’π‘ (πœ†, 𝐹𝑛,π‘š ) π‘’βˆ’π‘–πœ‹πœŽ(𝑁+1)π‘₯

βˆπ‘˜π‘ =1

π‘˜=1

+

(1 + πœ† βˆ’π‘ 

π‘’π‘–πœ‹πœŽπ‘₯ ) (1

+ πœ†π‘ 

π‘’βˆ’π‘–πœ‹πœŽπ‘₯ )

Let πœ† π‘˜ be chosen as in (38) and let π›Ύπ‘˜ (𝜏) be the coefficients of the following polynomial:

1

𝑝

3. Convergence Analysis

βˆπ‘ =1 (1 + πœ† βˆ’π‘  π‘’π‘–πœ‹πœŽπ‘₯ ) (1 + πœ† 𝑠 π‘’βˆ’π‘–πœ‹πœŽπ‘₯ )

Γ— βˆ‘ 𝛿𝑛𝑝 (πœ†, π‘“π‘›βˆ— βˆ’ 𝐹𝑛,π‘š ) π‘’π‘–πœ‹π‘›πœŽπ‘₯ |𝑛|≀𝑁

+ Γ—

βˆπ‘ =1 (1 + πœ† βˆ’π‘  π‘’π‘–πœ‹πœŽπ‘₯ ) (1 + πœ† 𝑠 π‘’βˆ’π‘–πœ‹πœŽπ‘₯ ) βˆ‘ 𝛿𝑛𝑝 |𝑛|>𝑁

where the first two terms can be assumed as corrections of the error. This observation leads to the following QPR interpolation:

π‘˜=0

(39)

𝜏1 𝜏 ) 𝑐 = Ξ”1𝑛 (𝑐𝑛 ) βˆ’ 1 π‘π‘›βˆ’1 . (40) 𝑁 π‘›βˆ’1 𝑁

For 𝑝 = 2, we have Ξ”2𝑛 (πœ†+ , 𝑐𝑛 ) = Ξ”1𝑛 (πœ†+ , 𝑐𝑛 ) + (1 βˆ’ = Ξ”1𝑛 (𝑐𝑛 ) βˆ’

𝜏2 ) Ξ”1π‘›βˆ’1 (πœ†+ , 𝑐𝑛 ) 𝑁

𝜏1 𝑐 𝑁 π‘›βˆ’1

𝜏2 𝜏 ) (Ξ”1π‘›βˆ’1 (𝑐𝑛 ) βˆ’ 1 π‘π‘›βˆ’2 ) 𝑁 𝑁 𝜏 + 𝜏2 1 𝜏𝜏 = Ξ”2𝑛 (𝑐𝑛 ) βˆ’ 1 Ξ” π‘›βˆ’1 (𝑐𝑛 ) + 1 22 Ξ”0π‘›βˆ’2 (𝑐𝑛 ) . 𝑁 𝑁 (41)

𝑝

𝐼𝑁,π‘š (𝑓, π‘₯)

+ (1 βˆ’

= 𝐼𝑁,π‘š (𝑓, π‘₯) 𝑝

+ βˆ‘ πœ† βˆ’π‘˜

π‘˜βˆ’1 π‘˜βˆ’1 π›Ώβˆ’π‘βˆ’1 (πœ†, 𝐹𝑛,π‘š ) π‘’βˆ’π‘–πœ‹πœŽπ‘π‘₯ βˆ’ 𝛿𝑁 (πœ†, 𝐹𝑛,π‘š ) π‘’π‘–πœ‹πœŽ(𝑁+1)π‘₯

βˆπ‘˜π‘ =1 (1 + πœ† βˆ’π‘  π‘’π‘–πœ‹πœŽπ‘₯ ) (1 + πœ† 𝑠 π‘’βˆ’π‘–πœ‹πœŽπ‘₯ )

π‘˜=1

+ βˆ‘πœ†π‘˜

𝑠=1

Ξ”1𝑛 (πœ†+ , 𝑐𝑛 ) = 𝑐𝑛 + (1 βˆ’

(πœ†, π‘“π‘›βˆ— ) π‘’π‘–πœ‹π‘›πœŽπ‘₯ , (35)

𝑝

𝑝

where 𝜏 = {𝜏1 , . . . , πœπ‘ }. Let us modify (20) in view of (38). For 𝑝 = 1, we write

1

𝑝

𝑝

∏ (1 + πœπ‘  π‘₯) = βˆ‘ π›Ύπ‘˜ (𝜏) π‘₯π‘˜ ,

π‘˜βˆ’1 𝛿𝑁+1

π‘–πœ‹πœŽπ‘π‘₯

(πœ†, 𝐹𝑛,π‘š ) 𝑒

π‘˜=1

βˆπ‘˜π‘ =1

(1 + πœ† βˆ’π‘ 

βˆ’

π‘˜βˆ’1 π›Ώβˆ’π‘

βˆ’π‘–πœ‹πœŽ(𝑁+1)π‘₯

(πœ†, 𝐹𝑛,π‘š ) 𝑒

π‘’π‘–πœ‹πœŽπ‘₯ ) (1

+ πœ† 𝑠 π‘’βˆ’π‘–πœ‹πœŽπ‘₯ )

(36)

In general, we can prove by the mathematical induction the following expansion [10]: 𝑝

𝑝

𝑠=0

𝑠=0

Δ𝑝𝑛 (πœ†+ , 𝑐𝑛 ) = βˆ‘π›Ύπ‘  (πœ†+ ) π‘π‘›βˆ’π‘  = βˆ‘(βˆ’1)𝑠

with the error 𝑝

𝑅𝑁,π‘š (𝑓, π‘₯) =

Now, let us modify (24) in view of (38). According to (20) and (42), we get (note that πœ†+ = πœ†βˆ’ )

1

𝑝

βˆπ‘ =1 (1 + πœ† βˆ’π‘  π‘’π‘–πœ‹πœŽπ‘₯ ) (1 + πœ† 𝑠 π‘’βˆ’π‘–πœ‹πœŽπ‘₯ )

+

𝑝

1

(37)

Γ— βˆ‘ 𝛿𝑛𝑝 (πœ†, π‘“π‘›βˆ— ) π‘’π‘–πœ‹πœŽπ‘›π‘₯ .

𝑠=0

𝑝

(43)

Similar to (42), we can show that

The QPR interpolation is undefined until parameters πœ† π‘˜ are unknown. Hence, determination of these parameters is a crucial problem for realization of the QPR interpolation. First, we assume that π‘˜ = 1, . . . , 𝑝,

π‘˜=0

𝛾 (𝜏) π‘βˆ’π‘  = βˆ‘(βˆ’1)𝑠 𝑠 𝑠 βˆ‘ π›Ύπ‘˜ (πœ†+ ) Ξ” 𝑛+π‘˜βˆ’π‘  (𝑐𝑛 ) . 𝑁 𝑠=0 π‘˜=0

|𝑛|>𝑁

πœ† βˆ’π‘˜

𝑝

𝑝

βˆπ‘ =1 (1 + πœ† βˆ’π‘  π‘’π‘–πœ‹πœŽπ‘₯ ) (1 + πœ† 𝑠 π‘’βˆ’π‘–πœ‹πœŽπ‘₯ )

𝜏 = πœ†π‘˜ = 1 βˆ’ π‘˜ , 𝑁

𝑝

𝛿𝑛𝑝 (πœ†, 𝑐𝑛 ) = βˆ‘ π›Ύπ‘˜ (πœ†+ ) βˆ‘π›Ύπ‘  (πœ†βˆ’ ) 𝑐𝑛+π‘˜βˆ’π‘ 

Γ— βˆ‘ 𝛿𝑛𝑝 (πœ†, π‘“π‘›βˆ— βˆ’ 𝐹𝑛,π‘š ) π‘’π‘–πœ‹πœŽπ‘›π‘₯ |𝑛|≀𝑁

𝛾𝑠 (𝜏) π‘βˆ’π‘  Ξ” (𝑐 ) . 𝑁𝑠 π‘›βˆ’π‘  𝑛 (42)

(38)

where πœπ‘˜ are some new parameters independent of 𝑁. In the next section, we investigate convergence of the QPR interpolation independent of the choice of parameters πœπ‘˜ . Then, we discuss some choices of these parameters. We also consider an approach connected with the idea of the FourierPade interpolation which leads to quasi-periodic FourierPade interpolation.

𝑝

𝑝

π‘˜=0

π‘˜=0

βˆ‘ π›Ύπ‘˜ (πœ†+ ) 𝑐𝑛+π‘˜ = βˆ‘ (βˆ’1)π‘˜

π›Ύπ‘˜ (𝜏) π‘βˆ’π‘˜ Ξ” (𝑐 ) . π‘π‘˜ 𝑛+𝑝 𝑛

(44)

Then, from (43), we have 𝑝

𝛿𝑛𝑝 (πœ†, 𝑐𝑛 ) = βˆ‘(βˆ’1)𝑠 𝑠=0

𝑝

𝛾𝑠 (𝜏) 𝛾 (𝜏) π‘βˆ’π‘  βˆ‘ (βˆ’1)π‘˜ π‘˜ π‘˜ Ξ”π‘βˆ’π‘˜ 𝑛+𝑝 (Ξ” π‘›βˆ’π‘  (𝑐𝑛 )) . 𝑁𝑠 π‘˜=0 𝑁 (45)

This leads to the following needed expansion: 𝑝

𝛿𝑛𝑝 (πœ†, 𝑐𝑛 ) = βˆ‘(βˆ’1)𝑠 𝑠=0

π‘βˆ’π‘˜

π‘βˆ’π‘ 

𝑝

𝛾𝑠 (𝜏) 𝛾 (𝜏) βˆ‘ (βˆ’1)π‘˜ π‘˜ π‘˜ Ξ”2π‘βˆ’π‘˜βˆ’π‘  𝑛+π‘βˆ’π‘  (𝑐𝑛 ) (46) 𝑠 𝑁 π‘˜=0 𝑁

2π‘βˆ’π‘˜βˆ’π‘ 

as Ξ” 𝑛+𝑝 (Ξ” π‘›βˆ’π‘  (𝑐𝑛 )) = Ξ” 𝑛+π‘βˆ’π‘  (𝑐𝑛 ).

International Journal of Analysis

5

Then,

and, consequently,

𝛿𝑛𝑀 (𝛿𝑛𝑝 (πœ†, 𝑐𝑛 )) 𝑝

= βˆ‘(βˆ’1)𝑠 𝑠=0

𝑝

𝛾𝑠 (𝜏) 𝛾 (𝜏) 2π‘βˆ’π‘˜βˆ’π‘  βˆ‘ (βˆ’1)π‘˜ π‘˜ π‘˜ Ξ”2𝑀 𝑛+𝑀 (Ξ” 𝑛+π‘βˆ’π‘  (𝑐𝑛 )) . 𝑁𝑠 π‘˜=0 𝑁 (47)

𝛿𝑛𝑀 (𝛿𝑛𝑝 (πœ†, π‘“π‘›βˆ— )) =

𝑠=0

𝛾𝑠 (𝜏) 𝛾 (𝜏) βˆ‘ (βˆ’1)π‘˜ π‘˜ π‘˜ Ξ”2𝑀+2π‘βˆ’π‘˜βˆ’π‘  𝑛+π‘βˆ’π‘ +𝑀 (𝑐𝑛 ) . 𝑁𝑠 π‘˜=0 𝑁

Γ— 𝛿𝑛𝑀 (𝛿𝑛𝑝 (πœ†, 𝐡𝑛 (π‘˜))) + π‘œ (π‘›βˆ’π‘žβˆ’2π‘βˆ’π‘šβˆ’1 ) . (55) Taking into account (49) and estimating Δ𝑝𝑛 (𝐡𝑛 (π‘˜)) 𝑂(π‘›βˆ’π‘˜βˆ’1βˆ’π‘ ) (see details in [9]), we get

(49)

1 1 𝑓𝑛 = ∫ 𝑓 (π‘₯) π‘’βˆ’π‘–πœ‹π‘›π‘₯ 𝑑π‘₯. 2 βˆ’1

𝛿𝑛𝑀 (𝛿𝑛𝑝 (πœ†, (βˆ’1)𝑛 π‘’π‘–πœ‹π›½π‘›/(2𝑁+π‘š+1) ))

(50)

=(

𝑝

(51)

π‘˜=0

(βˆ’1)π‘Ÿ(π‘š+1) . 2π‘βˆ’π‘˜βˆ’π‘ +𝑗+1 π‘Ÿ=βˆ’βˆž (2π‘Ÿ Β± 1)

𝑝

Lemma 3. Let 𝑓(π‘ž+2𝑝+π‘š) ∈ 𝐴𝐢[βˆ’1, 1] for some π‘ž β‰₯ 0, 𝑝, π‘š β‰₯ 1, and (βˆ’1) = 𝑓

(1) = 0,

π‘˜ = 0, . . . , π‘ž βˆ’ 1.

(52)

Let parameters πœ† π‘˜ be chosen as in (38). Then,

|𝑛| > 𝑁, 𝑁 󳨀→ ∞. (53) Proof. We have (see details in [5]) π‘ž+2𝑝+π‘š

βˆ‘ 𝑗=π‘ž

= βˆ‘(βˆ’1)𝑠 𝑠=0

𝑝

𝛾𝑠 (𝜏) 𝛾 (𝜏) βˆ‘ (βˆ’1)π‘˜ π‘˜ π‘˜ 𝑠 𝑁 π‘˜=0 𝑁

(59)

𝑛 π‘–πœ‹π›½π‘›/(2𝑁+π‘š+1) Γ— Ξ”2𝑀+2π‘βˆ’π‘˜βˆ’π‘  ). 𝑛+π‘βˆ’π‘ +𝑀 ((βˆ’1) 𝑒

In view of (27), we write Δ𝑝𝑛 ((βˆ’1)𝑛 π‘’π‘–πœ‹π›½π‘›/(2𝑁+π‘š+1) )

𝛿𝑛𝑀 (𝛿𝑛𝑝 (πœ†, π‘“π‘›βˆ— )) = 𝑂 (π‘›βˆ’2π‘€βˆ’1 π‘βˆ’2π‘βˆ’π‘ž ) + π‘œ (π‘›βˆ’π‘žβˆ’2π‘βˆ’π‘šβˆ’1 ) ,

π‘“π‘›βˆ— =

𝑝

πœ‹π›½ 2𝑝 2 π‘˜+𝑠 ) βˆ‘π›Ύπ‘  (𝜏) βˆ‘ (βˆ’1)π‘˜ π›Ύπ‘˜ (𝜏) ( ) . (58) 2 π‘–πœ‹π›½ 𝑠=0 π‘˜=0

𝛿𝑛𝑀 (𝛿𝑛𝑝 (πœ†, (βˆ’1)𝑛 π‘’π‘–πœ‹π›½π‘›/(2𝑁+π‘š+1) ))

First, we prove some lemmas.

𝑓

β„Žπ‘ (𝛽, 𝜏) = (

Proof. From (49), we have

Γ— βˆ‘

(π‘˜)

(57)

where

𝑝

∞

(π‘˜)

πœ‹π›½ 2𝑀 (βˆ’1)𝑛 π‘’π‘–πœ‹π›½π‘›/(2𝑁+π‘š+1) β„Žπ‘ (𝛽, 𝜏) ) 2 𝑁2𝑀+2𝑝

+ 𝑂 (π‘βˆ’2π‘€βˆ’2π‘βˆ’1 ) ,

𝑛 =ΜΈ 0, 𝐡0 (𝑗) = 0,

Β± πœ“π‘,π‘š,𝑗 (𝜏) = βˆ‘(βˆ’1)𝑠 𝛾𝑠 (𝜏) βˆ‘ π›Ύπ‘˜ (𝜏) (2𝑝 βˆ’ π‘˜ βˆ’ 𝑠 + 𝑗)!

(56)

which completes the proof.

Let (βˆ’1)𝑛+1 , 𝐡𝑛 (𝑗) = 2(π‘–πœ‹π‘›)𝑗+1

=

Lemma 4. Let πœ† π‘˜ be chosen as in (38), and let 𝛽 ∈ R be a constant. Then, the following estimate holds for 𝑛 ∈ Z as 𝑁 β†’ ∞:

We will frequently use the latest formula. We denote by 𝑓𝑛 the 𝑛th Fourier coefficient of 𝑓 as

𝑠=0

(𝑗 βˆ’ π‘˜)!

𝛿𝑛𝑀 (𝛿𝑛𝑝 (πœ†, 𝐡𝑛 (π‘˜))) = 𝑂 (π‘›βˆ’2π‘€βˆ’π‘˜βˆ’1 π‘βˆ’2𝑝 )

𝑝

𝑝

𝐴 π‘˜π‘— (𝑓) (π‘š + 1)π‘—βˆ’π‘˜ (2𝑁 + π‘š + 1)π‘˜

π‘˜=0

𝛿𝑛𝑀 (𝛿𝑛𝑝 (πœ†, 𝑐𝑛 )) = βˆ‘(βˆ’1)𝑠

1 2𝑗 𝑁𝑗

𝑗=π‘ž 𝑗

(48)

we find that

𝑝

βˆ‘

Γ—βˆ‘

Taking into account that 2π‘βˆ’π‘˜βˆ’π‘  2𝑀+2π‘βˆ’π‘˜βˆ’π‘  Ξ”2𝑀 𝑛+𝑀 (Ξ” 𝑛+π‘βˆ’π‘  (𝑐𝑛 )) = Ξ” 𝑛+π‘βˆ’π‘ +𝑀 (𝑐𝑛 ) ,

π‘ž+2𝑝+π‘š

𝑝

𝑝 = (βˆ’1) βˆ‘ ( ) (βˆ’1)π‘˜ π‘’π‘–πœ‹π›½(π‘›βˆ’π‘˜)/(2𝑁+π‘š+1) π‘˜ 𝑛

π‘˜=0

𝑝

𝑝 = (βˆ’1)𝑛 π‘’π‘–πœ‹π›½π‘›/(2𝑁+π‘š+1) βˆ‘ ( ) (βˆ’1)π‘˜ π‘˜

(60)

π‘˜=0

𝑗 𝐴 π‘˜π‘— (𝑓) (π‘š + 1)π‘—βˆ’π‘˜ (2𝑁 + π‘š + 1)π‘˜ 1 𝐡𝑛 (π‘˜) βˆ‘ 2𝑗 𝑁𝑗 π‘˜=0 (𝑗 βˆ’ π‘˜)!

∞

Γ—βˆ‘

𝑑

(π‘–πœ‹π›½) (βˆ’1)𝑑 π‘˜π‘‘

𝑑=0 𝑑!(2𝑁

+ π‘š + 1)𝑑

(54)

𝑑

(βˆ’1)𝑑 (π‘–πœ‹π›½) 𝑑 πœ”π‘,𝑑 , 𝑑=0 𝑑!(2𝑁 + π‘š + 1) ∞

+ π‘œ (π‘›βˆ’π‘žβˆ’2π‘βˆ’π‘šβˆ’1 ) ,

= (βˆ’1)𝑛 π‘’π‘–πœ‹π›½π‘›/(2𝑁+π‘š+1) βˆ‘

6

International Journal of Analysis

where

Then, in view of (54), we get 𝑝

𝑝 πœ”π‘,𝑑 = βˆ‘ ( ) (βˆ’1)𝑠 𝑠𝑑 . 𝑠 𝑠=0

(61)

βˆ— βˆ‘ 𝑓𝑛+π‘Ÿ(2𝑁+π‘š+1)

π‘Ÿ =ΜΈ 0

π‘ž+2𝑝+π‘š

=

Taking into account that πœ”π‘,𝑗 = 0, 0 ≀ 𝑗 < 𝑝 and πœ”π‘,𝑝 = (βˆ’1)𝑝 𝑝! (see [10], Lemma 2.1), we get

βˆ‘ 𝑗=π‘ž 𝑗

Γ—βˆ‘

Δ𝑝𝑛 ((βˆ’1)𝑛 π‘’π‘–πœ‹π›½π‘›/(2𝑁+π‘š+1) )

1 2𝑗 𝑁𝑗

𝐴 π‘˜π‘— (𝑓) (π‘š + 1)π‘—βˆ’π‘˜ (2𝑁 + π‘š + 1)π‘˜ (𝑗 βˆ’ π‘˜)!

π‘˜=0

𝑝

(βˆ’1)𝑛 (π‘–πœ‹π›½) π‘–πœ‹π›½π‘›/(2𝑁+π‘š+1) = 𝑒 + 𝑂 (π‘βˆ’π‘βˆ’1 ) . 2𝑝 𝑁𝑝

Γ— βˆ‘ 𝐡𝑛+π‘Ÿ(2𝑁+π‘š+1) (π‘˜) + π‘œ (π‘βˆ’π‘žβˆ’2π‘βˆ’π‘šβˆ’1 ) ,

(62)

π‘Ÿ =ΜΈ 0

π‘š

∞

β„“=1

π‘Ÿ=βˆ’βˆž

βˆ— βˆ‘ πœƒπ‘›,β„“ βˆ‘ 𝑓𝑁+β„“+π‘Ÿ(2𝑁+π‘š+1)

Then, 𝑛 π‘–πœ‹π›½π‘›/(2𝑁+π‘š+1) ) Ξ”2𝑀+2π‘βˆ’π‘˜βˆ’π‘  𝑛+π‘βˆ’π‘ +𝑀 ((βˆ’1) 𝑒

= (βˆ’1)𝑛+𝑝+𝑠+𝑀 (

=

π‘–πœ‹π›½ 2𝑀+2π‘βˆ’π‘˜βˆ’π‘  π‘–πœ‹π›½(𝑛+π‘βˆ’π‘ +𝑀)/(2𝑁+π‘š+1) 𝑒 ) 2𝑁

(βˆ’1)𝑛+1 2𝑁 + π‘š + 1 π‘ž+2𝑝+π‘š

Γ— βˆ‘ 𝑗=π‘ž

+ 𝑂 (π‘βˆ’2π‘€βˆ’2𝑝+π‘˜+π‘ βˆ’1 )

Γ— π‘’βˆ’π‘–πœ‹(π‘šβˆ’1)𝑛/(2𝑁+π‘š+1) βˆ‘ Ξ¦π‘˜,π‘š (𝑒2π‘–πœ‹(𝑁+β„“)/(2𝑁+π‘š+1) )

π‘–πœ‹π›½ 2𝑀+2π‘βˆ’π‘˜βˆ’π‘  ) 2𝑁

β„“=1

π‘š

βˆ’1 2πš€πœ‹π‘›(π‘ βˆ’1)/(2𝑁+π‘š+1) Γ— βˆ‘Vβ„“,𝑠 𝑒 + π‘œ (π‘βˆ’π‘žβˆ’2π‘βˆ’2 ) ,

+ 𝑂 (π‘βˆ’2π‘€βˆ’2𝑝+π‘˜+π‘ βˆ’1 )

𝑠=1

(63)

where we used estimate (16). According to the Taylor expansion

which completes the proof together with (59). Lemma 5. Let 𝑓(π‘ž+2𝑝+π‘š) ∈ 𝐴𝐢[βˆ’1, 1] for some π‘ž β‰₯ 0, 𝑝, π‘š β‰₯ 1, and 𝑓

(βˆ’1) = 𝑓

(π‘˜)

(1) = 0,

π‘˜ = 0, . . . , π‘ž βˆ’ 1.

Ξ¦π‘˜,π‘š (𝑒2πš€πœ‹(𝑁+β„“)/(2𝑁+π‘š+1) ) 2𝑝+π‘š

= βˆ‘

(64)

Let parameters πœ† π‘˜ be chosen as in (38). Then, the following estimate holds as 𝑁 β†’ ∞: 𝛿𝑛𝑀 (𝛿𝑛𝑝 (πœ†, 𝐹𝑛,π‘š βˆ’ π‘“π‘›βˆ— )) = 𝑂 (π‘βˆ’π‘žβˆ’2π‘€βˆ’2π‘βˆ’1 ) + π‘œ (π‘βˆ’π‘žβˆ’2π‘βˆ’2 ) , βˆ’ 𝑁 ≀ 𝑛 ≀ 𝑁 + π‘š. (65) Proof. First, we estimate 𝐹𝑛,π‘š βˆ’π‘“π‘›βˆ— . We have (see details in [5], at the beginning of the proof of Lemma 5)

𝑑=0

𝑑 1 (𝑑) Ξ¦π‘˜,π‘š (βˆ’1) (𝑒2π‘–πœ‹(𝑁+β„“)/(2𝑁+π‘š+1) + 1) 𝑑!

∞

π‘š

∞

π‘Ÿ=βˆ’βˆž

β„“=1

π‘Ÿ=βˆ’βˆž

π‘š

∞

β„“=1

π‘Ÿ=βˆ’βˆž

βˆ— βˆ— βˆ’ βˆ‘ πœƒπ‘›,β„“ βˆ‘ 𝑓𝑁+β„“+π‘Ÿ(2𝑁+π‘š+1) . 𝐹𝑛,π‘š βˆ’ π‘“π‘›βˆ— = βˆ‘ 𝑓𝑛+π‘Ÿ(2𝑁+π‘š+1)

(66)

(68)

+ 𝑂 (π‘βˆ’2π‘βˆ’π‘šβˆ’1 ) and relations π‘š

𝜏

π‘š

βˆ’1 2πš€πœ‹π‘›(π‘ βˆ’1)/(2𝑁+π‘š+1) 𝑒 βˆ‘ (𝑒2π‘–πœ‹(𝑁+β„“)/(2𝑁+π‘š+1) + 1) βˆ‘Vβ„“,𝑠 𝑠=1

β„“=1

2π‘–πœ‹π‘›/(2𝑁+π‘š+1)

= (𝑒

𝜏

+ 1) ,

𝜏 = 0, . . . , π‘š βˆ’ 1,

we derive 𝐹𝑛,π‘š βˆ’ π‘“π‘›βˆ— =

βˆ— βˆ— βˆ’ βˆ‘ πœƒπ‘›,β„“ βˆ‘ 𝑓𝑁+β„“+π‘Ÿ(2𝑁+π‘š+1) , 𝐹𝑛,π‘š = βˆ‘ 𝑓𝑛+π‘Ÿ(2𝑁+π‘š+1)

π‘Ÿ =ΜΈ 0

π‘—βˆ’π‘˜

𝑗

𝐴 π‘˜π‘— (𝑓) (π‘š + 1) 1 βˆ‘ 𝑗 𝑁 π‘˜=0 2π‘—βˆ’π‘˜ (π‘–πœ‹)π‘˜+1 (𝑗 βˆ’ π‘˜)! π‘š

= (βˆ’1)𝑛+𝑝+𝑠+𝑀 π‘’π‘–πœ‹π›½π‘›/(2𝑁+π‘š+1) (

(π‘˜)

(67)

π‘ž+2𝑝+π‘š

βˆ‘ 𝑗=π‘ž 𝑗

Γ—βˆ‘

1 2𝑗 𝑁𝑗

𝐴 π‘˜π‘— (𝑓) (π‘š + 1)π‘—βˆ’π‘˜ (2𝑁 + π‘š + 1)π‘˜

π‘˜=0

(𝑗 βˆ’ π‘˜)!

Γ— βˆ‘ 𝐡𝑛+π‘Ÿ(2𝑁+π‘š+1) (π‘˜) π‘Ÿ =ΜΈ 0

(69)

International Journal of Analysis +

7

(βˆ’1)𝑛 π‘’βˆ’π‘–πœ‹(π‘šβˆ’1)𝑛/(2𝑁+π‘š+1) 2𝑁 + π‘š + 1

Proof. We proceed as in the proof of Lemma 5 and derive

𝑗 𝐴 π‘˜π‘— (𝑓) (π‘š + 1)π‘—βˆ’π‘˜ 1 Γ— βˆ‘ βˆ‘ 𝑗 π‘—βˆ’π‘˜ (π‘–πœ‹)π‘˜+1 (𝑗 βˆ’ π‘˜)! 𝑗=π‘ž 𝑁 π‘˜=0 2 π‘ž+2𝑝+π‘š

π‘šβˆ’1 Ξ¦(𝑑) π‘˜,π‘š

Γ— [βˆ‘

(βˆ’1)

𝑑!

𝑑=0 π‘š+2𝑝

+ βˆ‘

(𝑒2π‘–πœ‹π‘›/(2𝑁+π‘š+1) + 1)

π‘Ÿ=βˆ’βˆž

π‘ž+2𝑝+π‘š

2π‘–πœ‹(𝑁+β„“)/(2𝑁+π‘š+1)

+

𝑑

+ 1)

𝑗

π‘šβˆ’1 Ξ¦(𝑑) π‘˜,π‘š

Γ— [βˆ‘

𝑝

π‘š+2𝑝

+ βˆ‘ π‘š

π‘š

βˆ’1 Γ— βˆ‘Vβ„“,𝑠 𝑠=1

(72)

𝑝

Γ— 𝛿±𝑁 (πœ†, (βˆ’1)𝑛 π‘’π‘–πœ‹π‘›(2π‘ βˆ’1βˆ’π‘š)/(2𝑁+π‘š+1) ) ]

Let parameters πœ† π‘˜ be chosen as in (38). Then, (βˆ’1)𝑁+1 2π‘ž+1 π‘π‘ž+2𝑝+1

+ π‘œ (π‘βˆ’π‘žβˆ’2π‘βˆ’2 ) . (76)

𝐴 π‘˜π‘ž (𝑓) (π‘š + 1)π‘žβˆ’π‘˜ 2π‘˜ (π‘ž βˆ’ π‘˜)!(π‘–πœ‹)π‘˜+1

π‘˜=0

Γ— [ (βˆ’1)𝑝

Β± πœ“π‘,π‘š,π‘˜ (𝜏)

π‘˜!

π‘šβˆ’1 Ξ¦

βˆ’ βˆ‘

𝑑=0

(𝑑) π‘˜,π‘š

This completes the proof in view of Lemma 4 and the following estimate: (βˆ’1)

𝑑!

∞

(73)

𝑀 𝛿±𝑁 (𝛿𝑛𝑝 (πœ†, βˆ‘ 𝐡𝑛+π‘Ÿ(2𝑁+π‘š+1) (𝑗))) π‘Ÿ=βˆ’βˆž

𝑑

𝑑 Γ— βˆ‘ ( ) π‘’Β±π‘–πœ‹π‘(2πœ‡βˆ’π‘š+1)/(2𝑁+π‘š+1) πœ‡ πœ‡=0

=

+ 𝑂 (π‘βˆ’π‘žβˆ’2π‘βˆ’2 ) , 𝑝

𝛿±(𝑁+1) (πœ†, 𝐹𝑛,π‘š ) = βˆ’π›ΏΒ±π‘ (πœ†, 𝐹𝑛,π‘š ) + 𝑂 (π‘βˆ’π‘žβˆ’2π‘βˆ’2 ) , 𝑝

𝑝

(βˆ’1)𝑁+𝑝+𝑀+1 Β± πœ“π‘,π‘š,𝑗+2𝑀 (𝜏) 2(π‘–πœ‹π‘)𝑗+1 𝑁2𝑀+2𝑝 𝑗!

(77)

+ 𝑂 (π‘βˆ’2π‘€βˆ’2π‘βˆ’π‘—βˆ’2 ) .

Γ— β„Žπ‘ (2πœ‡ βˆ’ π‘š + 1, 𝜏) ]

𝑝

𝑑

β„“=1

Lemma 6. Let 𝑓(π‘ž+2𝑝+π‘š) ∈ 𝐴𝐢[βˆ’1, 1] for some π‘ž β‰₯ 0, 𝑝, π‘š β‰₯ 1, and

Γ—βˆ‘

𝑑!

Γ— βˆ‘ (𝑒2π‘–πœ‹(𝑁+β„“)/(2𝑁+π‘š+1) + 1)

+1 = 𝑂(1/𝑁).

π‘˜ = 0, . . . , π‘ž βˆ’ 1.

Ξ¦(𝑑) π‘˜,π‘š (βˆ’1)

𝑑=π‘š

(71) We used also the fact that 𝑒

𝑑

𝑑 βˆ‘( ) πœ‡ πœ‡=0

Γ— 𝛿±𝑁 (πœ†, (βˆ’1)𝑛 π‘’π‘–πœ‹π‘›(2πœ‡βˆ’π‘š+1)/(2𝑁+π‘š+1) )

π‘Ÿ =ΜΈ 0

2π‘–πœ‹(𝑁+β„“)/(2𝑁+π‘š+1)

(𝑗 βˆ’ π‘˜)!

(βˆ’1)

𝑑!

𝑑=0

𝛿𝑛𝑀 (𝛿𝑛𝑝 (πœ†, βˆ‘ 𝐡𝑛+π‘Ÿ(2𝑁+π‘š+1) (π‘˜))) = 𝑂 (π‘βˆ’2π‘€βˆ’2π‘βˆ’π‘˜βˆ’1 ) .

(1) = 0,

𝐴 π‘˜π‘— (𝑓) (π‘š + 1)π‘—βˆ’π‘˜

π‘—βˆ’π‘˜ (π‘–πœ‹)π‘˜+1 π‘˜=0 2

This completes the proof in view of (49), Lemma 4, and the following estimate [9]:

π‘ž

1 1 βˆ‘ 2𝑁 + π‘š + 1 𝑗=π‘ž 𝑁𝑗

Γ—βˆ‘

(70)

𝑝

∞

𝑑!

+ π‘œ (π‘βˆ’π‘žβˆ’2π‘βˆ’2 ) .

𝛿±𝑁 (πœ†, 𝐹𝑛,π‘š ) =

(𝑗 βˆ’ π‘˜)!(π‘š + 1)π‘˜βˆ’π‘—

𝑝

βˆ’1 2πš€πœ‹π‘›(π‘ βˆ’1)/(2𝑁+π‘š+1) Γ—βˆ‘Vβ„“,𝑠 𝑒 ] 𝑠=1

(βˆ’1) = 𝑓

𝐴 π‘˜π‘— (𝑓) (2𝑁 + π‘š + 1)π‘˜

Γ— 𝛿±𝑁 (πœ†, βˆ‘ 𝐡𝑛+π‘Ÿ(2𝑁+π‘š+1) (π‘˜))

π‘š

𝑓

1 2𝑗 𝑁𝑗

π‘˜=0

β„“=1

(π‘˜)

𝑗=π‘ž 𝑗

Γ— βˆ‘ (𝑒

(π‘˜)

βˆ‘

Γ—βˆ‘

𝑑

Ξ¦(𝑑) π‘˜,π‘š (βˆ’1)

𝑑=π‘š π‘š

π‘ž+2𝑝+π‘š

𝑝

𝛿±𝑁 (πœ†, 𝐹𝑛,π‘š ) =

(74)

π›Ώβˆ’π‘ (πœ†, 𝐹𝑛,π‘š ) = (βˆ’1)π‘š+1 𝛿𝑁 (πœ†, 𝐹𝑛,π‘š ) + 𝑂 (π‘βˆ’π‘žβˆ’2π‘βˆ’2 ) . (75)

The proof of (77), for π‘š = 0, can be found in [9]. General case can be proved similarly and we omit it. Estimates (74) and (75) can be proved similarly. Now, we present the main results of the paper.

8

International Journal of Analysis

Theorem 7. Let 𝑓(π‘ž+2𝑝+π‘š) ∈ 𝐴𝐢[βˆ’1, 1] for some π‘ž β‰₯ 0, 𝑝, π‘š β‰₯ 1, and 𝑓(π‘˜) (βˆ’1) = 𝑓(π‘˜) (1) = 0,

π‘˜ = 0, . . . , π‘ž βˆ’ 1.

(78)

Let parameters πœ† π‘˜ be chosen as in (38). Then, the following estimate holds for |π‘₯| < 1: 𝑝

𝑅𝑁,π‘š (𝑓, π‘₯) =

(βˆ’1)𝑁 sin (πœ‹πœŽ (𝑁 + (1/2)) π‘₯ βˆ’ (πœ‹π‘š/2)) cos2𝑝+1 (πœ‹π‘₯/2) (2𝑁)π‘ž+2𝑝+1 π‘ž

Γ—βˆ‘

𝐴 π‘˜π‘ž (𝑓) (π‘š + 1)π‘žβˆ’π‘˜ 2π‘˜

π‘˜=0

Hence, the last term in the right-hand side of (80) is π‘œ(π‘βˆ’π‘žβˆ’2π‘βˆ’1 ). Then, by Lemma 5, 𝛿𝑛1 (𝛿𝑛𝑝 (πœ†, 𝐹𝑛,π‘š βˆ’ π‘“π‘›βˆ— )) = 𝑂 (π‘βˆ’π‘žβˆ’2π‘βˆ’3 ) + π‘œ (π‘βˆ’π‘žβˆ’2π‘βˆ’2 ) , βˆ’ 𝑁 ≀ 𝑛 ≀ 𝑁 + π‘š, and the fifth term is also π‘œ(π‘βˆ’π‘žβˆ’2π‘βˆ’1 ). Therefore, 𝑝

𝑝

𝑐 (π‘₯) 𝑅𝑁,π‘š (𝑓, π‘₯) = π‘’βˆ’π‘–πœ‹πœŽπ‘π‘₯ π›Ώβˆ’π‘βˆ’1 (πœ†, 𝐹𝑛,π‘š )

(π‘ž βˆ’ π‘˜)!π‘–π‘˜βˆ’π‘š πœ‹π‘˜+1

𝑝

βˆ’ π‘’π‘–πœ‹πœŽ(𝑁+1)π‘₯ 𝛿𝑁 (πœ†, 𝐹𝑛,π‘š )

πœ“+ (𝜏) π‘šβˆ’1 Ξ¦(𝑑) π‘˜,π‘š (βˆ’1) 𝑝 𝑝,π‘š,π‘˜ Γ— ((βˆ’1) βˆ’βˆ‘ π‘˜! 𝑑! 𝑑=0

𝑝

+ π‘’π‘–πœ‹πœŽπ‘π‘₯ 𝛿𝑁+1 (πœ†, 𝐹𝑛,π‘š )

𝑁 󳨀→ ∞. (79)

Proof. We have from (37) by the Abel transformation (see transformation from (33) to (34) with πœ† ≑ 1) 𝑝 βˆ’π‘–πœ‹πœŽπ‘π‘₯ π›Ώβˆ’π‘βˆ’1

𝑝

(πœ†, 𝐹𝑛,π‘š ) 𝑐 (π‘₯)

𝑅𝑁,π‘š (𝑓, π‘₯) = 𝑒

𝑝

βˆ’π‘’

(πœ†, 𝐹𝑛,π‘š ) 𝑐 (π‘₯)

(80)

𝑝

π›Ώβˆ’π‘ (πœ†, 𝐹𝑛,π‘š ) 𝑐 (π‘₯)

+

1 𝑁 1 𝑝 βˆ‘ 𝛿 (𝛿 (πœ†, π‘“π‘›βˆ— βˆ’ 𝐹𝑛,π‘š )) π‘’π‘–πœ‹πœŽπ‘›π‘₯ 𝑐 (π‘₯) 𝑛=βˆ’π‘ 𝑛 𝑛

+

1 βˆ‘ 𝛿1 (𝛿𝑝 (πœ†, π‘“π‘›βˆ— )) π‘’π‘–πœ‹πœŽπ‘›π‘₯ , 𝑐 (π‘₯) |𝑛|>𝑁 𝑛 𝑛

where 𝑝

πœ‹πœŽπ‘₯ 𝑐 (π‘₯) = 4cos ( ) ∏ (1 + πœ† βˆ’π‘  π‘’π‘–πœ‹πœŽπ‘₯ ) (1 + πœ† 𝑠 π‘’βˆ’π‘–πœ‹πœŽπ‘₯ ) . 2 𝑠=1 (81) 2

It is easy to verify that lim 𝑐 (π‘₯) = 22𝑝+2 cos2𝑝+2 (

π‘β†’βˆž

πœ‹π‘₯ ). 2

(82)

According to Lemma 3, 𝛿𝑛1 (𝛿𝑛𝑝 (πœ†, π‘“π‘›βˆ— )) = 𝑂 (π‘›βˆ’3 π‘βˆ’2π‘βˆ’π‘ž ) + π‘œ (π‘›βˆ’π‘žβˆ’2π‘βˆ’2 ) , |𝑛| > 𝑁,

𝑝

𝑝

𝑐 (π‘₯) 𝑅𝑁,π‘š (𝑓, π‘₯) = 𝛿𝑁 (πœ†, 𝐹𝑛,π‘š ) Γ— ((βˆ’1)π‘š (π‘’βˆ’π‘–πœ‹πœŽπ‘π‘₯ + π‘’βˆ’π‘–πœ‹πœŽ(𝑁+1)π‘₯ ) (86)

which concludes the proof in view of Lemma 6.

𝑝 π‘–πœ‹πœŽπ‘π‘₯ 𝛿𝑁+1

βˆ’ π‘’βˆ’π‘–πœ‹πœŽ(𝑁+1)π‘₯

Taking into account estimates (74) and (75), we get

+ π‘œ (π‘βˆ’π‘žβˆ’2π‘βˆ’1 )

𝑐 (π‘₯)

+𝑒

+ π‘œ (π‘βˆ’π‘žβˆ’2π‘βˆ’1 ) .

βˆ’ (π‘’π‘–πœ‹πœŽπ‘π‘₯ + π‘’π‘–πœ‹πœŽ(𝑁+1)π‘₯ ))

𝑝 π‘–πœ‹πœŽ(𝑁+1)π‘₯ 𝛿𝑁 (πœ†, 𝐹𝑛,π‘š )

(85)

βˆ’ π‘’βˆ’π‘–πœ‹πœŽ(𝑁+1)π‘₯ π›Ώβˆ’π‘ (πœ†, 𝐹𝑛,π‘š )

𝑑

𝑑 Γ— βˆ‘ ( ) 𝑖2πœ‡βˆ’π‘š+1 β„Žπ‘ (2πœ‡ βˆ’ π‘š + 1, 𝜏)) πœ‡ πœ‡=0 + π‘œ (π‘βˆ’π‘žβˆ’2π‘βˆ’1 ) ,

𝑁 󳨀→ ∞ (84)

𝑁 󳨀→ ∞.

(83)

Let us compare the results of Theorems 1 and 7. Theorem 1 investigates the pointwise convergence of the QP interpolation on (βˆ’1, 1) and states that for 𝑓(π‘ž+2π‘š) ∈ 𝐴𝐢[βˆ’1, 1] the convergence rate is 𝑂(π‘βˆ’π‘žβˆ’π‘šβˆ’1 ) for π‘š β‰₯ 1. Theorem 7 explores the pointwise convergence of the QPR interpolation and shows that convergence rate is 𝑂(π‘βˆ’π‘žβˆ’2π‘βˆ’1 ) for 𝑓(π‘ž+2𝑝+π‘š) ∈ 𝐴𝐢[βˆ’1, 1] and π‘š β‰₯ 1. We see that for π‘š = 2𝑝 both theorems are provided with the same rates of convergence by putting the same smoothness requirements on 𝑓, although the exact constants of the asymptotic errors are different. Then, we see that for 𝑝 > π‘š/2 the QPR interpolation has improved accuracy compared to the QP interpolation and improvement is by factor 𝑂(𝑁2π‘βˆ’π‘š ). In this case, Theorem 7 puts additional smoothness requirement on 𝑓 and comparison is valid if only the interpolated function has enough smoothness (for example, if it is infinitely differentiable). It is worth recalling that parameter π‘š indicates the size of the Vandermonde matrix (12) that must be inverted for realization of the QP and QPR interpolations. It is wellknown that the Vandermonde matrices are ill-conditioned and standard numerical methods fail to accurately compute the entries of the inverses when the sizes of the matrices are big. Hence, from practical point of view, it is more reasonable to take π‘š small (π‘š ≀ 6) and additional accuracy obtain by increasing 𝑝.

International Journal of Analysis

9 Γ—10βˆ’25

Γ—10βˆ’25 8

1.5

6

1

4 0.5

2

βˆ’0.6

βˆ’0.4

βˆ’0.2

0.2

0.4

0.6

βˆ’0.6

βˆ’0.4

βˆ’0.2

(a)

Figure 1: The graphs of the absolute errors (94)) (b) while interpolating (90).

𝑝

(87)

Another choice is based on the asymptotic estimates of Theorems 7 and 8. If it is possible to vanish the following expressions by the choice of parameters πœπ‘˜ : (βˆ’1)𝑝

Similarly, the case π‘š = 0 can be analyzed. We present the corresponding theorem without the proof which can be performed as the above one.

π‘˜ = 0, . . . , π‘ž βˆ’ 1.

sin (πœ‹π‘π‘₯) (βˆ’1)𝑁+𝑝 π‘ž+2𝑝+1 cos2𝑝+1 (πœ‹π‘₯/2) (2𝑁) 𝐴 π‘˜π‘ž (𝑓) 2π‘˜

π‘ž

Γ—βˆ‘ π‘˜=0

π‘–π‘˜ πœ‹π‘˜+1

+ π‘œ (𝑁

(π‘ž βˆ’ π‘˜)!π‘˜!

βˆ’π‘žβˆ’2π‘βˆ’1

+ πœ“π‘,0,π‘˜ (𝜏)

π‘˜!

π‘šβˆ’1 Ξ¦(𝑑) π‘˜,π‘š

βˆ’βˆ‘

𝑑=0

(βˆ’1)

𝑑!

𝑑

or + πœ“π‘,0,π‘˜ (𝜏) = 0,

π‘˜ = 0, . . . , π‘ž,

(89)

𝑝

𝑅𝑁,π‘š (𝑓, π‘₯) = π‘œ (π‘βˆ’π‘žβˆ’2π‘βˆ’1 ) ,

|π‘₯| < 1.

(93)

For example, when 𝑝 = π‘ž = 4 and π‘š = 2, we find 𝜏1 = βˆ’0.3047909 βˆ’ 0.4539705𝑖,

Comparison with Theorem 2 shows improvement by factor 𝑂(𝑁2𝑝 ) for any 𝑝 β‰₯ 1 if 𝑓 has enough smoothness.

𝜏2 = 0.9640011 βˆ’ 0.1481304𝑖, 𝜏3 = 6.375385 + 0.000739𝑖,

4. Parameter Determination in Rational Corrections

(94)

𝜏4 = 12.877189 βˆ’ 0.000022𝑖.

Till now, we did not discuss the problem of parameters πœπ‘˜ determination as Theorems 7 and 8 are valid for all choices. Now, let us consider some choices with the corresponding numerical results. Let 4

(92)

then, Theorems 7 and 8 will be provided with improved estimate

).

𝑓 (π‘₯) = (1 βˆ’ π‘₯2 ) sin (π‘₯ βˆ’ 1) .

(91)

π‘˜ = 0, . . . , π‘ž

(88)

Let parameters πœ† π‘˜ be chosen as in (38). Then, the following estimate holds for |π‘₯| < 1 as 𝑁 β†’ ∞: 𝑝

+ πœ“π‘,π‘š,π‘˜ (𝜏)

𝑑 Γ— βˆ‘ ( ) 𝑖2πœ‡βˆ’π‘š+1 β„Žπ‘ (2πœ‡ βˆ’ π‘š + 1, 𝜏) = 0, πœ‡ πœ‡=0

Theorem 8. Let 𝑓(π‘ž+2𝑝+1) ∈ 𝐴𝐢[βˆ’1, 1] for some π‘ž β‰₯ 0, 𝑝 β‰₯ 1, and

𝑅𝑁,0 (𝑓, π‘₯) =

0.6

on interval [βˆ’0.7, 0.7] for π‘š = 2 and 𝑝 = 4 when πœπ‘˜ = π‘˜ (a) and with optimal πœπ‘˜ (see

𝑁 󳨀→ ∞, π‘ž = 0.

𝑓(π‘˜) (βˆ’1) = 𝑓(π‘˜) (1) = 0,

0.4

(b) 𝑝 |𝑅256,π‘š (𝑓, π‘₯)|

Note that for π‘š = 1 the second term in the brackets of + (1, 𝜏) = 0. Hence, estimate (79) vanishes and also πœ“π‘,1,0 𝑅𝑁,1 (𝑓, π‘₯) = π‘œ (π‘βˆ’2π‘βˆ’1 ) ,

0.2

(90)

One choice is πœπ‘˜ = π‘˜ which shows satisfactory numerical results (see Figure 1).

4 With this choice 𝑅256,2 (𝑓, π‘₯) = π‘œ(π‘βˆ’13 ) for (90). Figure 1 compares the choices πœπ‘˜ = π‘˜ (a) and optimal choice (104) (b). As it was expected the precision of interpolation is higher in (b) compared to (a). The third choice is not connected with (38) and allows determining parameters πœ† π‘˜ immediately along the ideas of the Fourier-Pade interpolation ([11]). This approach is more complex as πœ† π‘˜ must be recalculated for each function 𝑓 and for each 𝑁 and as a consequence leads to nonlinear interpolation but, however, is much more precise when |π‘₯| < 1.

10

International Journal of Analysis Γ—10βˆ’35 8

[3] 6

[4]

4 2

[5] 0.7

βˆ’0.7

𝑝 Figure 2: The graph of |𝑅256,π‘š (𝑓, π‘₯)| on interval [βˆ’0.7, 0.7] for π‘š

[6]

=2 and 𝑝 = 4 while interpolating (90) by the quasi-periodic FourierPade interpolation (see (96)). [7]

More specifically, parameters πœ† can be determined from the following system: 𝛿𝑛𝑝 (πœ†, 𝐹𝑛,π‘š ) = 0,

|𝑛| = 𝑁 βˆ’ 𝑝 + 1, . . . , 𝑁.

(95)

[8]

We will refer to this interpolation as quasi-periodic FourierPade interpolation. For example, when π‘š = 2, 𝑝 = 4, and 𝑁 = 256, then, for (90), we get

[9]

πœ† 1 = 0.922 + 0.010𝑖,

πœ† 2 = 0.952 + 0.010𝑖,

πœ† 3 = 0.975 + 0.010𝑖,

πœ† 4 = 0.994 βˆ’ 0.011𝑖,

πœ† βˆ’1 = 0.994 + 0.011𝑖,

πœ† βˆ’2 = 0.975 βˆ’ 0.010𝑖,

πœ† βˆ’3 = 0.952 βˆ’ 0.010𝑖,

πœ† βˆ’4 = 0.922 βˆ’ 0.010𝑖.

[10]

(96)

4 Figure 2 shows the graph of |𝑅256,2 (𝑓, π‘₯)| for function (90) with πœ† π‘˜ from (96). Comparison with Figure 1 shows high precision of the quasi-periodic Fourier-Pade interpolation for |π‘₯| < 1 compared to other choices of parameters πœ† π‘˜ . Theoretical analysis of convergence of the quasi-periodic Fourier-Pade interpolation will be carried out elsewhere.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments The authors acknowledge the reviewers and the editors for their valuable suggestions and constructive comments that helped to improve the paper.

References [1] A. Nersessian and N. Hovhannisyan, β€œQuasiperiodic interpolation,” Reports of the National Academy of Sciences of Armenia, vol. 101, no. 2, pp. 115–121, 2001. [2] A. Nersessian and N. Hovhannisyan, β€œMinimization of errors of the polynomial-trigonometric interpolation with shifted

[11]

nodes,” 2007, Preprint of the Institute of Mathematics of the National Academy of Sciences of Armenia, http://math.sci .am/upload/preprints/toPrprt.pdf. L. Poghosyan, β€œOn a convergence of the quasi-periodic interpolation,” in Proceedings of the 2012 International Workshop on Functional Analysis, Timisoara, Romania, October 2012. L. Poghosyan, β€œOn L2 -convergence of the quasi-periodic interpolation,” Reports of the National Academy of Sciences of Armenia, vol. 113, no. 3, pp. 240–247, 2013. L. Poghosyan and A. Poghosyan, β€œAsymptotic estimates for the quasi-periodic interpolations,” Armenian Journal of Mathematics, vol. 5, no. 1, pp. 34–57, 2013. L. Poghosyan and A. Poghosyan, β€œOn a pointwise convergence of the quasi-periodictrigonometric interpolation,” Proceedings of the National Academy of Sciences of Armeniaβ€”Mathematics. In press. L. Poghosyan and A. Poghosyan, β€œConvergence acceleration of the quasi-periodic interpolation by rational and polynomial corrections (abstract),” in Proceedings of the 2nd International Conference Mathematics in Armenia: Advances and Perspectives, Tsaghkadzor, Armenia, August 2013. I. Gohberg and V. Olshevsky, β€œThe fast generalized Parker-Traub algorithm for inversion of Vandermonde and related matrices,” Journal of Complexity, vol. 13, no. 2, pp. 208–234, 1997. A. Poghosyan, β€œOn a fast convergence of the rationaltrigonometric-polynomial interpolation,” Advances in Numerical Analysis, vol. 2013, Article ID 315748, 13 pages, 2013. A. Nersessian and A. Poghosyan, β€œOn a rational linear approximation of Fourier series for smooth functions,” Journal of Scientific Computing, vol. 26, no. 1, pp. 111–125, 2006. G. A. Baker and P. Graves-Morris, Pade Approximants, Encyclopedia of Mathematics and Its Applications, vol. 59, Cambridge University Press, Cambridge, UK, 2nd edition, 1966.

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