Research Article On a Pointwise Convergence of

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 © 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,𝑚 (𝑓, 𝑥), 𝑚 ≥ 0 (𝑚 is integer) and 𝑥 ∈ [−1, 1], interpolates function 𝑓 on equidistant grid 𝑘 𝑥𝑘 = , 𝑁

|𝑛| ≤ 𝑁,

𝑘 = 0, . . . , 𝑞,

∞

|𝑘| ≤ 𝑁

(1)

and is exact for a quasi-periodic function 𝑒𝑖𝜋𝑛𝜎𝑥 ,

𝐴 𝑠𝑘 (𝑓) = 𝑓(𝑘) (1) − (−1)𝑘+𝑠 𝑓(𝑘) (−1) ,

2𝑁 𝜎= 2𝑁 + 𝑚 + 1

(2)

with period 2/𝜎 which tends to 2 as 𝑁 → ∞. 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 (−1, 1)-norm and at the endpoints 𝑥 = ±1 in terms of the limit function. Pointwise convergence in the interval (−1, 1) is explored in [6]. The main results there, which we need for further comparison, are the following theorems.

(−1)𝑟(𝑚+1) . 𝑘+1 𝑟=−∞ (2𝑟 + 𝑥)

Φ𝑘,𝑚 (𝑒𝑖𝜋𝑥 ) = 𝑒(𝑖𝜋/2)(𝑚−1)𝑥 ∑

(3)

We denote by 𝑅𝑁,𝑚 (𝑓, 𝑥) the error of the QP interpolation as follows:

𝑅𝑁,𝑚 (𝑓, 𝑥) = 𝑓 (𝑥) − 𝐼𝑁,𝑚 (𝑓, 𝑥) .

(4)

Theorem 1 (see [6]). Let 𝑓(𝑞+2𝑚) ∈ 𝐴𝐶[−1, 1] for some 𝑚 ≥ 1, 𝑞 ≥ 0, and

𝑓(𝑘) (−1) = 𝑓(𝑘) (1) = 0,

𝑘 = 0, . . . , 𝑞 − 1.

(5)

2

International Journal of Analysis

Then, the following estimate holds for |𝑥| < 1 as 𝑁 → ∞:

where 𝑚

𝑅𝑁,𝑚 (𝑓, 𝑥) = 𝐶𝑞,𝑚 (𝑓)

̌ − ∑𝜃 𝑓̌ 𝐹𝑛,𝑚 = 𝑓𝑛,𝑚 𝑛,ℓ ℓ+𝑁,𝑚 ,

𝑁

(−1) 𝑁𝑞+𝑚+1

ℓ=1

̌ = 𝑓𝑛,𝑚

[𝑚/2]

𝑚−𝑘 × [sin (𝜋 (𝑁 + 1) 𝜎𝑥) ∑ ( ) 𝑘 𝑘=0

×

(11)

𝑚

−1 2𝑖𝜋𝑛(𝑠−1)/(2𝑁+𝑚+1) 𝜃𝑛,ℓ = 𝑒2𝑖𝜋(ℓ+𝑁−𝑛)(𝑁+𝑚)/(2𝑁+𝑚+1) ∑Vℓ,𝑠 𝑒 .

𝑘

(−1) − sin (𝜋𝑁𝜎𝑥) 22𝑘+1 cos2𝑘+2 (𝜋𝑥/2)

𝑠=1

−1 Here, Vℓ,𝑠

are the elements of the inverse of the Vandermonde matrix (V𝑠,ℓ ) as

[𝑚/2]−1

𝑚−𝑘−2 × ∑ ( ) 𝑘 𝑘=0

V𝑠,ℓ = 𝛼ℓ𝑠−1 ,

𝑘

(−1) ] + 𝑜 (𝑁−𝑞−𝑚−1 ) , 22𝑘+3 cos2𝑘+4 (𝜋𝑥/2) (6)

×

𝑁 1 𝑘 ∑ 𝑓 ( ) 𝑒−2𝑖𝜋𝑛𝑘/(2𝑁+𝑚+1) , 2𝑁 + 𝑚 + 1 𝑘=−𝑁 𝑁

where

𝛼ℓ = 𝑒2𝑖𝜋(ℓ+𝑁)/(2𝑁+𝑚+1) ,

(12)

and have the following explicit form [8]: −1 =− Vℓ,𝑠

1

𝛼ℓ𝑠 ∏𝑚 𝑖=1, 𝑖 ≠ ℓ

𝑠−1

𝑗

∑𝛽 𝛼 , (𝛼ℓ − 𝛼𝑖 ) 𝑗=0 𝑗 ℓ

ℓ, 𝑠 = 1, . . . , 𝑚, (13)

where 𝛽𝑗 are the coefficients of the following polynomial: 𝑞

𝐶𝑞,𝑚 (𝑓) = ∑

𝐴 𝑘𝑞 (𝑓) (𝑚 + 1)𝑞−𝑘

𝑞−𝑘+1 𝑖𝑘−1 𝜋𝑘−𝑚+1 𝑘=0 2

Theorem 2 (see [6]). Let 𝑓 and

(𝑞+1)

(𝑞 − 𝑘)!

Φ(𝑚) 𝑘,𝑚

(−1) .

(7)

∈ 𝐴𝐶[−1, 1] for some 𝑞 ≥ 0

𝑓(𝑘) (−1) = 𝑓(𝑘) (1) = 0,

𝑘 = 0, . . . , 𝑞 − 1.

(8)

Then, the following estimate holds for |𝑥| < 1 as 𝑁 → ∞: 𝑅𝑁,0 (𝑓, 𝑥) = 𝐴 0𝑞 (𝑓) [𝑞/2]

× ∑ 𝑘=0

(9)

In the current paper, we consider convergence acceleration of the QP interpolation by rational corrections in terms of 𝑒𝑖𝜋𝜎𝑥 which leads to quasi-periodic-rational (QPR) interpolation. We investigate the pointwise convergence of the QPR interpolation in the interval (−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=−𝑁

𝑗=1

𝑗=0

get

(14)

Taking into account that 𝛼𝑠 − 𝛼𝑖 = 𝑂(1/𝑁), from (13), we −1 Vℓ,𝑠 = 𝑂 (𝑁𝑚−1 ) ,

𝑁 󳨀→ ∞,

(15)

𝜃𝑛,ℓ = 𝑂 (𝑁𝑚−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 𝜇 = {𝜇𝑘 }𝑘=1 . By Δ𝑘𝑛 (𝜇, 𝑐𝑛 ), we denote generalized finite differences defined by the following recurrent relations: Δ0𝑛 (𝜇, 𝑐𝑛 ) = 𝑐𝑛 ,

+ 𝑜 (𝑁−𝑞−1 ) .

𝐼𝑁,𝑚 (𝑓, 𝑥) = ∑ 𝐹𝑛,𝑚 𝑒𝑖𝜋𝑛𝜎𝑥 ,

𝑚

2. Quasi-Periodic-Rational Interpolation

(−1)𝑁 sin (𝜋𝑁𝑥) 2𝑞+1 𝑁𝑞+1 cos (𝜋𝑥/2)

∞ (−1)𝑘 22𝑘 (−1)𝑠 ∑ (𝑞 − 2𝑘)!𝜋2𝑘+1 𝑠=−∞ (2𝑠 + 1)2𝑘+1

𝑚

∏ (𝑥 − 𝛼𝑗 ) = ∑ 𝛽𝑗 𝑥𝑗 .

𝑚 ∈ 𝑍, 𝑚 ≥ 0,

(10)

𝑘−1 Δ𝑘𝑛 (𝜇, 𝑐𝑛 ) = Δ𝑘−1 𝑛 (𝜇, 𝑐𝑛 ) + 𝜇𝑘 Δ 𝑛−1 (𝜇, 𝑐𝑛 ) ,

𝑘 ≥ 1,

(17)

for some sequence 𝑐𝑛 . When 𝜇 ≡ 1, we put Δ𝑘𝑛 (𝜇, 𝑐𝑛 ) = Δ𝑘𝑛 (𝑐𝑛 ) .

(18)

It is easy to verify that Δ1𝑛 (𝜇, 𝑐𝑛 ) = 𝑐𝑛 + 𝜇1 𝑐𝑛−1 , Δ2𝑛 (𝜇, 𝑐𝑛 ) = Δ1𝑛 (𝜇, 𝑐𝑛 ) + 𝜇2 Δ1𝑛−1 (𝜇, 𝑐𝑛 ) = 𝑐𝑛 + (𝜇1 + 𝜇2 ) 𝑐𝑛−1 + 𝜇1 𝜇2 𝑐𝑛−2 , Δ3𝑛 (𝜇, 𝑐𝑛 ) = 𝑐𝑛 + (𝜇1 + 𝜇2 + 𝜇3 ) 𝑐𝑛−1 + (𝜇1 𝜇2 + 𝜇1 𝜇3 + 𝜇2 𝜇3 ) 𝑐𝑛−2 + 𝜇1 𝜇2 𝜇3 𝑐𝑛−3 . (19)


3

In general, we can prove by the mathematical induction that

where

𝑝

Δ𝑝𝑛 (𝜇, 𝑐𝑛 ) = ∑𝛾𝑠 (𝜇) 𝑐𝑛−𝑠 ,

𝛼

𝑗 𝑓(𝑗) (−1) 𝑥 ( + 1) , 𝑗! 𝜎 𝑗=0

(20)

𝑓left (𝑥) = ∑

𝑠=0

where 𝛾𝑠 (𝜇) are the coefficients of the following polynomial: 𝑝

𝑝

𝑠=0

𝑠=1

∑𝛾𝑠 (𝜇) 𝑥𝑠 = ∏ (1 + 𝜇𝑠 𝑥) .

𝛼

𝑗 𝑓(𝑗) (1) 𝑥 𝑓right (𝑥) = ∑ ( − 1) . 𝑗! 𝜎 𝑗=0

(21) 𝑝 {𝜆 𝑘 }|𝑘|=1 ,

+

According to definition of 𝑓∗ , we can write

Consider the following vectors: 𝜆 = 𝜆 = 𝑝 𝑝 {𝜆 𝑘 }𝑘=1 , and 𝜆− = {𝜆 −𝑘 }𝑘=1 . By 𝛿𝑛𝑘 (𝜆, 𝑐𝑛 ), we denote modified finite differences defined by the following recurrent relations: 𝛿𝑛0 (𝜆, 𝑐𝑛 ) = 𝑐𝑛 , 𝛿𝑛𝑘

(𝜆, 𝑐𝑛 ) =

𝛿𝑛𝑘−1

(𝜆, 𝑐𝑛 ) +

𝑘−1 𝜆 −𝑘 𝛿𝑛−1

∞

𝑓∗ (𝑥) = ∑ 𝑓𝑛∗ 𝑒𝑖𝜋𝑛𝑥 , 𝑛=−∞

∞

𝑓 (𝑥) = ∑ 𝑓𝑛∗ 𝑒𝑖𝜋𝑛𝜎𝑥 ,

𝑘 ≥ 1, (22)

𝑛=−∞

(23)

𝑝

𝑝

(24)

𝑠=0

where

𝑁

𝑝

𝑝

𝑘=1

𝑘=0

𝑝

𝑝

𝑘=1

𝑘=0

|𝑛|>𝑁

The following transformation is easy to verify (see details in [9] for similar transformation): 𝑅𝑁,𝑚 (𝑓, 𝑥) = 𝜆 −1

∏ (1 + 𝜆 𝑘 𝑥) = ∑ 𝛾𝑘 (𝜆+ ) 𝑥𝑘 ,

𝐹−𝑁−1,𝑚 𝑒−𝑖𝜋𝜎𝑁𝑥 − 𝐹𝑁,𝑚 𝑒𝑖𝜋𝜎(𝑁+1)𝑥 (1 + 𝜆 −1 𝑒𝑖𝜋𝜎𝑥 ) (1 + 𝜆 1 𝑒−𝑖𝜋𝜎𝑥 )

+ 𝜆1

(25)

∏ (1 + 𝜆 −𝑘 𝑥) = ∑ 𝛾𝑘 (𝜆− ) 𝑥𝑘 . +

It is easy to verify that

𝐹𝑁+1,𝑚 𝑒𝑖𝜋𝜎𝑁𝑥 − 𝐹−𝑁,𝑚 𝑒−𝑖𝜋𝜎(𝑁+1)𝑥 (1 + 𝜆 −1 𝑒𝑖𝜋𝜎𝑥 ) (1 + 𝜆 1 𝑒−𝑖𝜋𝜎𝑥 ) 1

(1 + 𝜆 −1

𝑒𝑖𝜋𝜎𝑥 ) (1

+ 𝜆 1 𝑒−𝑖𝜋𝜎𝑥 )

(34)

𝑁

𝛿𝑛𝑘 (𝑐𝑛 ) = Δ2𝑘 𝑛+𝑘 (𝑐𝑛 ) ,

(26)

× ∑ 𝛿𝑛1 (𝜆, 𝑓𝑛∗ − 𝐹𝑛,𝑚 ) 𝑒𝑖𝜋𝑛𝜎𝑥

(27)

+

𝑛=−𝑁

𝑘

𝑘 Δ𝑘𝑛 (𝑐𝑛 ) = ∑ ( ) 𝑐𝑛−𝑠 , 𝑠 𝑠=0 2𝑘 (𝑐𝑛 ) = ∑ ( ) 𝑐𝑛+𝑘−𝑠 . 𝑠 𝑠=0

𝑓left (𝑥) , 𝑥 ∈ [−1, −𝜎) , { { { 𝑥 𝑥 ∈ [−𝜎, 𝜎] , 𝑓 (𝑥) = {𝑓 ( ) , { { 𝜎 {𝑓right (𝑥) , 𝑥 ∈ (𝜎, 1] ,

1 (1 + 𝜆 −1 𝑒𝑖𝜋𝜎𝑥 ) (1 + 𝜆 1 𝑒−𝑖𝜋𝜎𝑥 )

× ∑ 𝛿𝑛1 (𝜆, 𝑓𝑛∗ ) 𝑒𝑖𝜋𝑛𝜎𝑥 .

2𝑘

(28)

We assume that 𝑓 ∈ 𝐶𝛼 [−1, 1] for some 𝛼 ≥ 1 and we denote ∗

(32)

𝑛=−𝑁

𝛿𝑛𝑝 (𝜆, 𝑐𝑛 ) = ∑ 𝛾𝑘 (𝜆+ ) ∑𝛾𝑠 (𝜆− ) 𝑐𝑛+𝑘−𝑠 ,

𝛿𝑛𝑘

𝑥 ∈ [−1, 1] .

𝑅𝑁,𝑚 (𝑓, 𝑥) = ∑ (𝑓𝑛∗ − 𝐹𝑛,𝑚 ) 𝑒𝑖𝜋𝑛𝜎𝑥 + ∑ 𝑓𝑛∗ 𝑒𝑖𝜋𝑛𝜎𝑥 . (33)

Similar to (20), we can show that

𝑘=0

(31)

Therefore,

for some sequence 𝑐𝑛 . When 𝜆 ≡ 1, we put 𝛿𝑛𝑘 (𝜆, 𝑐𝑛 ) = 𝛿𝑛𝑘 (𝑐𝑛 ) .

𝑥 ∈ (−1, 1) .

Hence,

(𝜆, 𝑐𝑛 )

𝑘−1 (𝜆, 𝑐𝑛 ) + 𝜆 −𝑘 𝛿𝑛𝑘−1 (𝜆, 𝑐𝑛 )) , + 𝜆 𝑘 (𝛿𝑛+1

(30)

|𝑛|>𝑁

Reiteration of it up to 𝑝 times leads to the following expansion of the error: 𝑅𝑁,𝑚 (𝑓, 𝑥)

(29)

𝑝

= ∑ 𝜆 −𝑘 𝑘=1

𝑘−1 𝑘−1 𝛿−𝑁−1 (𝜆, 𝐹𝑛,𝑚 ) 𝑒−𝑖𝜋𝜎𝑁𝑥 − 𝛿𝑁 (𝜆, 𝐹𝑛,𝑚 ) 𝑒𝑖𝜋𝜎(𝑁+1)𝑥

∏𝑘𝑠=1 (1 + 𝜆 −𝑠 𝑒𝑖𝜋𝜎𝑥 ) (1 + 𝜆 𝑠 𝑒−𝑖𝜋𝜎𝑥 )

4

International Journal of Analysis 𝑝

+ ∑𝜆𝑘

𝑘−1 𝑘−1 𝛿𝑁+1 (𝜆, 𝐹𝑛,𝑚 ) 𝑒𝑖𝜋𝜎𝑁𝑥 − 𝛿−𝑁 (𝜆, 𝐹𝑛,𝑚 ) 𝑒−𝑖𝜋𝜎(𝑁+1)𝑥

∏𝑘𝑠=1

𝑘=1

+

(1 + 𝜆 −𝑠


+ 𝜆𝑠

𝑒−𝑖𝜋𝜎𝑥 )

Let 𝜆 𝑘 be chosen as in (38) and let 𝛾𝑘 (𝜏) be the coefficients of the following polynomial:

1

𝑝

3. Convergence Analysis

∏𝑠=1 (1 + 𝜆 −𝑠 𝑒𝑖𝜋𝜎𝑥 ) (1 + 𝜆 𝑠 𝑒−𝑖𝜋𝜎𝑥 )

× ∑ 𝛿𝑛𝑝 (𝜆, 𝑓𝑛∗ − 𝐹𝑛,𝑚 ) 𝑒𝑖𝜋𝑛𝜎𝑥 |𝑛|≤𝑁

+ ×

∏𝑠=1 (1 + 𝜆 −𝑠 𝑒𝑖𝜋𝜎𝑥 ) (1 + 𝜆 𝑠 𝑒−𝑖𝜋𝜎𝑥 ) ∑ 𝛿𝑛𝑝 |𝑛|>𝑁

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

𝑘=0

(39)

𝜏1 𝜏 ) 𝑐 = Δ1𝑛 (𝑐𝑛 ) − 1 𝑐𝑛−1 . (40) 𝑁 𝑛−1 𝑁

For 𝑝 = 2, we have Δ2𝑛 (𝜆+ , 𝑐𝑛 ) = Δ1𝑛 (𝜆+ , 𝑐𝑛 ) + (1 − = Δ1𝑛 (𝑐𝑛 ) −

𝜏2 ) Δ1𝑛−1 (𝜆+ , 𝑐𝑛 ) 𝑁

𝜏1 𝑐 𝑁 𝑛−1

𝜏2 𝜏 ) (Δ1𝑛−1 (𝑐𝑛 ) − 1 𝑐𝑛−2 ) 𝑁 𝑁 𝜏 + 𝜏2 1 𝜏𝜏 = Δ2𝑛 (𝑐𝑛 ) − 1 Δ 𝑛−1 (𝑐𝑛 ) + 1 22 Δ0𝑛−2 (𝑐𝑛 ) . 𝑁 𝑁 (41)

𝑝

𝐼𝑁,𝑚 (𝑓, 𝑥)

+ (1 −

= 𝐼𝑁,𝑚 (𝑓, 𝑥) 𝑝

+ ∑ 𝜆 −𝑘

𝑘−1 𝑘−1 𝛿−𝑁−1 (𝜆, 𝐹𝑛,𝑚 ) 𝑒−𝑖𝜋𝜎𝑁𝑥 − 𝛿𝑁 (𝜆, 𝐹𝑛,𝑚 ) 𝑒𝑖𝜋𝜎(𝑁+1)𝑥

∏𝑘𝑠=1 (1 + 𝜆 −𝑠 𝑒𝑖𝜋𝜎𝑥 ) (1 + 𝜆 𝑠 𝑒−𝑖𝜋𝜎𝑥 )

𝑘=1

+ ∑𝜆𝑘

𝑠=1

Δ1𝑛 (𝜆+ , 𝑐𝑛 ) = 𝑐𝑛 + (1 −

(𝜆, 𝑓𝑛∗ ) 𝑒𝑖𝜋𝑛𝜎𝑥 , (35)

𝑝

𝑝

where 𝜏 = {𝜏1 , . . . , 𝜏𝑝 }. Let us modify (20) in view of (38). For 𝑝 = 1, we write

1

𝑝

𝑝

∏ (1 + 𝜏𝑠 𝑥) = ∑ 𝛾𝑘 (𝜏) 𝑥𝑘 ,

𝑘−1 𝛿𝑁+1

𝑖𝜋𝜎𝑁𝑥

(𝜆, 𝐹𝑛,𝑚 ) 𝑒

𝑘=1

∏𝑘𝑠=1

(1 + 𝜆 −𝑠

−

𝑘−1 𝛿−𝑁

−𝑖𝜋𝜎(𝑁+1)𝑥

(𝜆, 𝐹𝑛,𝑚 ) 𝑒


+ 𝜆 𝑠 𝑒−𝑖𝜋𝜎𝑥 )

(36)

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

𝑝

𝑠=0

𝑠=0

Δ𝑝𝑛 (𝜆+ , 𝑐𝑛 ) = ∑𝛾𝑠 (𝜆+ ) 𝑐𝑛−𝑠 = ∑(−1)𝑠

with the error 𝑝

𝑅𝑁,𝑚 (𝑓, 𝑥) =

Now, let us modify (24) in view of (38). According to (20) and (42), we get (note that 𝜆+ = 𝜆− )

1

𝑝


+

𝑝

1

(37)

× ∑ 𝛿𝑛𝑝 (𝜆, 𝑓𝑛∗ ) 𝑒𝑖𝜋𝜎𝑛𝑥 .

𝑠=0

𝑝

(43)

Similar to (42), we can show that

The QPR interpolation is undefined until parameters 𝜆 𝑘 are unknown. Hence, determination of these parameters is a crucial problem for realization of the QPR interpolation. First, we assume that 𝑘 = 1, . . . , 𝑝,

𝑘=0

𝛾 (𝜏) 𝑝−𝑠 = ∑(−1)𝑠 𝑠 𝑠 ∑ 𝛾𝑘 (𝜆+ ) Δ 𝑛+𝑘−𝑠 (𝑐𝑛 ) . 𝑁 𝑠=0 𝑘=0

|𝑛|>𝑁

𝜆 −𝑘

𝑝

𝑝


𝜏 = 𝜆𝑘 = 1 − 𝑘 , 𝑁

𝑝

𝛿𝑛𝑝 (𝜆, 𝑐𝑛 ) = ∑ 𝛾𝑘 (𝜆+ ) ∑𝛾𝑠 (𝜆− ) 𝑐𝑛+𝑘−𝑠

× ∑ 𝛿𝑛𝑝 (𝜆, 𝑓𝑛∗ − 𝐹𝑛,𝑚 ) 𝑒𝑖𝜋𝜎𝑛𝑥 |𝑛|≤𝑁

𝛾𝑠 (𝜏) 𝑝−𝑠 Δ (𝑐 ) . 𝑁𝑠 𝑛−𝑠 𝑛 (42)

(38)

where 𝜏𝑘 are some new parameters independent of 𝑁. In the next section, we investigate convergence of the QPR interpolation independent of the choice of parameters 𝜏𝑘 . 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

𝑘=0

𝑘=0

∑ 𝛾𝑘 (𝜆+ ) 𝑐𝑛+𝑘 = ∑ (−1)𝑘

𝛾𝑘 (𝜏) 𝑝−𝑘 Δ (𝑐 ) . 𝑁𝑘 𝑛+𝑝 𝑛

(44)

Then, from (43), we have 𝑝

𝛿𝑛𝑝 (𝜆, 𝑐𝑛 ) = ∑(−1)𝑠 𝑠=0

𝑝

𝛾𝑠 (𝜏) 𝛾 (𝜏) 𝑝−𝑠 ∑ (−1)𝑘 𝑘 𝑘 Δ𝑝−𝑘 𝑛+𝑝 (Δ 𝑛−𝑠 (𝑐𝑛 )) . 𝑁𝑠 𝑘=0 𝑁 (45)

This leads to the following needed expansion: 𝑝

𝛿𝑛𝑝 (𝜆, 𝑐𝑛 ) = ∑(−1)𝑠 𝑠=0

𝑝−𝑘

𝑝−𝑠

𝑝

𝛾𝑠 (𝜏) 𝛾 (𝜏) ∑ (−1)𝑘 𝑘 𝑘 Δ2𝑝−𝑘−𝑠 𝑛+𝑝−𝑠 (𝑐𝑛 ) (46) 𝑠 𝑁 𝑘=0 𝑁

2𝑝−𝑘−𝑠

as Δ 𝑛+𝑝 (Δ 𝑛−𝑠 (𝑐𝑛 )) = Δ 𝑛+𝑝−𝑠 (𝑐𝑛 ).


5

Then,

and, consequently,

𝛿𝑛𝑤 (𝛿𝑛𝑝 (𝜆, 𝑐𝑛 )) 𝑝

= ∑(−1)𝑠 𝑠=0

𝑝

𝛾𝑠 (𝜏) 𝛾 (𝜏) 2𝑝−𝑘−𝑠 ∑ (−1)𝑘 𝑘 𝑘 Δ2𝑤 𝑛+𝑤 (Δ 𝑛+𝑝−𝑠 (𝑐𝑛 )) . 𝑁𝑠 𝑘=0 𝑁 (47)

𝛿𝑛𝑤 (𝛿𝑛𝑝 (𝜆, 𝑓𝑛∗ )) =

𝑠=0

𝛾𝑠 (𝜏) 𝛾 (𝜏) ∑ (−1)𝑘 𝑘 𝑘 Δ2𝑤+2𝑝−𝑘−𝑠 𝑛+𝑝−𝑠+𝑤 (𝑐𝑛 ) . 𝑁𝑠 𝑘=0 𝑁

× 𝛿𝑛𝑤 (𝛿𝑛𝑝 (𝜆, 𝐵𝑛 (𝑘))) + 𝑜 (𝑛−𝑞−2𝑝−𝑚−1 ) . (55) Taking into account (49) and estimating Δ𝑝𝑛 (𝐵𝑛 (𝑘)) 𝑂(𝑛−𝑘−1−𝑝 ) (see details in [9]), we get

(49)

1 1 𝑓𝑛 = ∫ 𝑓 (𝑥) 𝑒−𝑖𝜋𝑛𝑥 𝑑𝑥. 2 −1

𝛿𝑛𝑤 (𝛿𝑛𝑝 (𝜆, (−1)𝑛 𝑒𝑖𝜋𝛽𝑛/(2𝑁+𝑚+1) ))

(50)

=(

𝑝

(51)

𝑘=0

(−1)𝑟(𝑚+1) . 2𝑝−𝑘−𝑠+𝑗+1 𝑟=−∞ (2𝑟 ± 1)

𝑝

Lemma 3. Let 𝑓(𝑞+2𝑝+𝑚) ∈ 𝐴𝐶[−1, 1] for some 𝑞 ≥ 0, 𝑝, 𝑚 ≥ 1, and (−1) = 𝑓

(1) = 0,

𝑘 = 0, . . . , 𝑞 − 1.

(52)

Let parameters 𝜆 𝑘 be chosen as in (38). Then,

|𝑛| > 𝑁, 𝑁 󳨀→ ∞. (53) Proof. We have (see details in [5]) 𝑞+2𝑝+𝑚

∑ 𝑗=𝑞

= ∑(−1)𝑠 𝑠=0

𝑝

𝛾𝑠 (𝜏) 𝛾 (𝜏) ∑ (−1)𝑘 𝑘 𝑘 𝑠 𝑁 𝑘=0 𝑁

(59)

𝑛 𝑖𝜋𝛽𝑛/(2𝑁+𝑚+1) × Δ2𝑤+2𝑝−𝑘−𝑠 ). 𝑛+𝑝−𝑠+𝑤 ((−1) 𝑒

In view of (27), we write Δ𝑝𝑛 ((−1)𝑛 𝑒𝑖𝜋𝛽𝑛/(2𝑁+𝑚+1) )

𝛿𝑛𝑤 (𝛿𝑛𝑝 (𝜆, 𝑓𝑛∗ )) = 𝑂 (𝑛−2𝑤−1 𝑁−2𝑝−𝑞 ) + 𝑜 (𝑛−𝑞−2𝑝−𝑚−1 ) ,

𝑓𝑛∗ =

𝑝

𝜋𝛽 2𝑝 2 𝑘+𝑠 ) ∑𝛾𝑠 (𝜏) ∑ (−1)𝑘 𝛾𝑘 (𝜏) ( ) . (58) 2 𝑖𝜋𝛽 𝑠=0 𝑘=0

𝛿𝑛𝑤 (𝛿𝑛𝑝 (𝜆, (−1)𝑛 𝑒𝑖𝜋𝛽𝑛/(2𝑁+𝑚+1) ))

First, we prove some lemmas.

𝑓

ℎ𝑝 (𝛽, 𝜏) = (

Proof. From (49), we have

× ∑

(𝑘)

(57)

where

𝑝

∞

(𝑘)

𝜋𝛽 2𝑤 (−1)𝑛 𝑒𝑖𝜋𝛽𝑛/(2𝑁+𝑚+1) ℎ𝑝 (𝛽, 𝜏) ) 2 𝑁2𝑤+2𝑝

+ 𝑂 (𝑁−2𝑤−2𝑝−1 ) ,

𝑛 ≠ 0, 𝐵0 (𝑗) = 0,

± 𝜓𝑝,𝑚,𝑗 (𝜏) = ∑(−1)𝑠 𝛾𝑠 (𝜏) ∑ 𝛾𝑘 (𝜏) (2𝑝 − 𝑘 − 𝑠 + 𝑗)!

(56)

which completes the proof.

Let (−1)𝑛+1 , 𝐵𝑛 (𝑗) = 2(𝑖𝜋𝑛)𝑗+1

=

Lemma 4. Let 𝜆 𝑘 be chosen as in (38), and let 𝛽 ∈ R be a constant. Then, the following estimate holds for 𝑛 ∈ Z as 𝑁 → ∞:

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

𝑠=0

(𝑗 − 𝑘)!

𝛿𝑛𝑤 (𝛿𝑛𝑝 (𝜆, 𝐵𝑛 (𝑘))) = 𝑂 (𝑛−2𝑤−𝑘−1 𝑁−2𝑝 )

𝑝

𝑝

𝐴 𝑘𝑗 (𝑓) (𝑚 + 1)𝑗−𝑘 (2𝑁 + 𝑚 + 1)𝑘

𝑘=0

𝛿𝑛𝑤 (𝛿𝑛𝑝 (𝜆, 𝑐𝑛 )) = ∑(−1)𝑠

1 2𝑗 𝑁𝑗

𝑗=𝑞 𝑗

(48)

we find that

𝑝

∑

×∑

Taking into account that 2𝑝−𝑘−𝑠 2𝑤+2𝑝−𝑘−𝑠 Δ2𝑤 𝑛+𝑤 (Δ 𝑛+𝑝−𝑠 (𝑐𝑛 )) = Δ 𝑛+𝑝−𝑠+𝑤 (𝑐𝑛 ) ,

𝑞+2𝑝+𝑚

𝑝

𝑝 = (−1) ∑ ( ) (−1)𝑘 𝑒𝑖𝜋𝛽(𝑛−𝑘)/(2𝑁+𝑚+1) 𝑘 𝑛

𝑘=0

𝑝

𝑝 = (−1)𝑛 𝑒𝑖𝜋𝛽𝑛/(2𝑁+𝑚+1) ∑ ( ) (−1)𝑘 𝑘

(60)

𝑘=0

𝑗 𝐴 𝑘𝑗 (𝑓) (𝑚 + 1)𝑗−𝑘 (2𝑁 + 𝑚 + 1)𝑘 1 𝐵𝑛 (𝑘) ∑ 2𝑗 𝑁𝑗 𝑘=0 (𝑗 − 𝑘)!

∞

×∑

𝑡

(𝑖𝜋𝛽) (−1)𝑡 𝑘𝑡

𝑡=0 𝑡!(2𝑁

+ 𝑚 + 1)𝑡

(54)

𝑡

(−1)𝑡 (𝑖𝜋𝛽) 𝑡 𝜔𝑝,𝑡 , 𝑡=0 𝑡!(2𝑁 + 𝑚 + 1) ∞

+ 𝑜 (𝑛−𝑞−2𝑝−𝑚−1 ) ,

= (−1)𝑛 𝑒𝑖𝜋𝛽𝑛/(2𝑁+𝑚+1) ∑

6


where

Then, in view of (54), we get 𝑝

𝑝 𝜔𝑝,𝑡 = ∑ ( ) (−1)𝑠 𝑠𝑡 . 𝑠 𝑠=0

(61)

∗ ∑ 𝑓𝑛+𝑟(2𝑁+𝑚+1)

𝑟 ≠ 0

𝑞+2𝑝+𝑚

=

Taking into account that 𝜔𝑝,𝑗 = 0, 0 ≤ 𝑗 < 𝑝 and 𝜔𝑝,𝑝 = (−1)𝑝 𝑝! (see [10], Lemma 2.1), we get

∑ 𝑗=𝑞 𝑗

×∑

Δ𝑝𝑛 ((−1)𝑛 𝑒𝑖𝜋𝛽𝑛/(2𝑁+𝑚+1) )

1 2𝑗 𝑁𝑗

𝐴 𝑘𝑗 (𝑓) (𝑚 + 1)𝑗−𝑘 (2𝑁 + 𝑚 + 1)𝑘 (𝑗 − 𝑘)!

𝑘=0

𝑝

(−1)𝑛 (𝑖𝜋𝛽) 𝑖𝜋𝛽𝑛/(2𝑁+𝑚+1) = 𝑒 + 𝑂 (𝑁−𝑝−1 ) . 2𝑝 𝑁𝑝

× ∑ 𝐵𝑛+𝑟(2𝑁+𝑚+1) (𝑘) + 𝑜 (𝑁−𝑞−2𝑝−𝑚−1 ) ,

(62)

𝑟 ≠ 0

𝑚

∞

ℓ=1

𝑟=−∞

∗ ∑ 𝜃𝑛,ℓ ∑ 𝑓𝑁+ℓ+𝑟(2𝑁+𝑚+1)

Then, 𝑛 𝑖𝜋𝛽𝑛/(2𝑁+𝑚+1) ) Δ2𝑤+2𝑝−𝑘−𝑠 𝑛+𝑝−𝑠+𝑤 ((−1) 𝑒

= (−1)𝑛+𝑝+𝑠+𝑤 (

=

𝑖𝜋𝛽 2𝑤+2𝑝−𝑘−𝑠 𝑖𝜋𝛽(𝑛+𝑝−𝑠+𝑤)/(2𝑁+𝑚+1) 𝑒 ) 2𝑁

(−1)𝑛+1 2𝑁 + 𝑚 + 1 𝑞+2𝑝+𝑚

× ∑ 𝑗=𝑞

+ 𝑂 (𝑁−2𝑤−2𝑝+𝑘+𝑠−1 )

× 𝑒−𝑖𝜋(𝑚−1)𝑛/(2𝑁+𝑚+1) ∑ Φ𝑘,𝑚 (𝑒2𝑖𝜋(𝑁+ℓ)/(2𝑁+𝑚+1) )

𝑖𝜋𝛽 2𝑤+2𝑝−𝑘−𝑠 ) 2𝑁

ℓ=1

𝑚

−1 2𝚤𝜋𝑛(𝑠−1)/(2𝑁+𝑚+1) × ∑Vℓ,𝑠 𝑒 + 𝑜 (𝑁−𝑞−2𝑝−2 ) ,

+ 𝑂 (𝑁−2𝑤−2𝑝+𝑘+𝑠−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(𝑞+2𝑝+𝑚) ∈ 𝐴𝐶[−1, 1] for some 𝑞 ≥ 0, 𝑝, 𝑚 ≥ 1, and 𝑓

(−1) = 𝑓

(𝑘)

(1) = 0,

𝑘 = 0, . . . , 𝑞 − 1.

Φ𝑘,𝑚 (𝑒2𝚤𝜋(𝑁+ℓ)/(2𝑁+𝑚+1) ) 2𝑝+𝑚

= ∑

(64)

Let parameters 𝜆 𝑘 be chosen as in (38). Then, the following estimate holds as 𝑁 → ∞: 𝛿𝑛𝑤 (𝛿𝑛𝑝 (𝜆, 𝐹𝑛,𝑚 − 𝑓𝑛∗ )) = 𝑂 (𝑁−𝑞−2𝑤−2𝑝−1 ) + 𝑜 (𝑁−𝑞−2𝑝−2 ) , − 𝑁 ≤ 𝑛 ≤ 𝑁 + 𝑚. (65) Proof. First, we estimate 𝐹𝑛,𝑚 −𝑓𝑛∗ . We have (see details in [5], at the beginning of the proof of Lemma 5)

𝑡=0

𝑡 1 (𝑡) Φ𝑘,𝑚 (−1) (𝑒2𝑖𝜋(𝑁+ℓ)/(2𝑁+𝑚+1) + 1) 𝑡!

∞

𝑚

∞

𝑟=−∞

ℓ=1

𝑟=−∞

𝑚

∞

ℓ=1

𝑟=−∞

∗ ∗ − ∑ 𝜃𝑛,ℓ ∑ 𝑓𝑁+ℓ+𝑟(2𝑁+𝑚+1) . 𝐹𝑛,𝑚 − 𝑓𝑛∗ = ∑ 𝑓𝑛+𝑟(2𝑁+𝑚+1)

(66)

(68)

+ 𝑂 (𝑁−2𝑝−𝑚−1 ) and relations 𝑚

𝜏

𝑚

−1 2𝚤𝜋𝑛(𝑠−1)/(2𝑁+𝑚+1) 𝑒 ∑ (𝑒2𝑖𝜋(𝑁+ℓ)/(2𝑁+𝑚+1) + 1) ∑Vℓ,𝑠 𝑠=1

ℓ=1

2𝑖𝜋𝑛/(2𝑁+𝑚+1)

= (𝑒

𝜏

+ 1) ,

𝜏 = 0, . . . , 𝑚 − 1,

we derive 𝐹𝑛,𝑚 − 𝑓𝑛∗ =

∗ ∗ − ∑ 𝜃𝑛,ℓ ∑ 𝑓𝑁+ℓ+𝑟(2𝑁+𝑚+1) , 𝐹𝑛,𝑚 = ∑ 𝑓𝑛+𝑟(2𝑁+𝑚+1)

𝑟 ≠ 0

𝑗−𝑘

𝑗

𝐴 𝑘𝑗 (𝑓) (𝑚 + 1) 1 ∑ 𝑗 𝑁 𝑘=0 2𝑗−𝑘 (𝑖𝜋)𝑘+1 (𝑗 − 𝑘)! 𝑚

= (−1)𝑛+𝑝+𝑠+𝑤 𝑒𝑖𝜋𝛽𝑛/(2𝑁+𝑚+1) (

(𝑘)

(67)

𝑞+2𝑝+𝑚

∑ 𝑗=𝑞 𝑗

×∑

1 2𝑗 𝑁𝑗

𝐴 𝑘𝑗 (𝑓) (𝑚 + 1)𝑗−𝑘 (2𝑁 + 𝑚 + 1)𝑘

𝑘=0

(𝑗 − 𝑘)!

× ∑ 𝐵𝑛+𝑟(2𝑁+𝑚+1) (𝑘) 𝑟 ≠ 0

(69)

International Journal of Analysis +

7

(−1)𝑛 𝑒−𝑖𝜋(𝑚−1)𝑛/(2𝑁+𝑚+1) 2𝑁 + 𝑚 + 1

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

𝑗 𝐴 𝑘𝑗 (𝑓) (𝑚 + 1)𝑗−𝑘 1 × ∑ ∑ 𝑗 𝑗−𝑘 (𝑖𝜋)𝑘+1 (𝑗 − 𝑘)! 𝑗=𝑞 𝑁 𝑘=0 2 𝑞+2𝑝+𝑚

𝑚−1 Φ(𝑡) 𝑘,𝑚

× [∑

(−1)

𝑡!

𝑡=0 𝑚+2𝑝

+ ∑

(𝑒2𝑖𝜋𝑛/(2𝑁+𝑚+1) + 1)

𝑟=−∞

𝑞+2𝑝+𝑚

2𝑖𝜋(𝑁+ℓ)/(2𝑁+𝑚+1)

+

𝑡

+ 1)

𝑗


× [∑

𝑝

𝑚+2𝑝

+ ∑ 𝑚

𝑚

−1 × ∑Vℓ,𝑠 𝑠=1

(72)

𝑝

× 𝛿±𝑁 (𝜆, (−1)𝑛 𝑒𝑖𝜋𝑛(2𝑠−1−𝑚)/(2𝑁+𝑚+1) ) ]

Let parameters 𝜆 𝑘 be chosen as in (38). Then, (−1)𝑁+1 2𝑞+1 𝑁𝑞+2𝑝+1

+ 𝑜 (𝑁−𝑞−2𝑝−2 ) . (76)

𝐴 𝑘𝑞 (𝑓) (𝑚 + 1)𝑞−𝑘 2𝑘 (𝑞 − 𝑘)!(𝑖𝜋)𝑘+1

𝑘=0

× [ (−1)𝑝

± 𝜓𝑝,𝑚,𝑘 (𝜏)

𝑘!

𝑚−1 Φ

− ∑

𝑡=0

(𝑡) 𝑘,𝑚

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

𝑡!

∞

(73)

𝑤 𝛿±𝑁 (𝛿𝑛𝑝 (𝜆, ∑ 𝐵𝑛+𝑟(2𝑁+𝑚+1) (𝑗))) 𝑟=−∞

𝑡

𝑡 × ∑ ( ) 𝑒±𝑖𝜋𝑁(2𝜇−𝑚+1)/(2𝑁+𝑚+1) 𝜇 𝜇=0

=

+ 𝑂 (𝑁−𝑞−2𝑝−2 ) , 𝑝

𝛿±(𝑁+1) (𝜆, 𝐹𝑛,𝑚 ) = −𝛿±𝑁 (𝜆, 𝐹𝑛,𝑚 ) + 𝑂 (𝑁−𝑞−2𝑝−2 ) , 𝑝

𝑝

(−1)𝑁+𝑝+𝑤+1 ± 𝜓𝑝,𝑚,𝑗+2𝑤 (𝜏) 2(𝑖𝜋𝑁)𝑗+1 𝑁2𝑤+2𝑝 𝑗!

(77)

+ 𝑂 (𝑁−2𝑤−2𝑝−𝑗−2 ) .

× ℎ𝑝 (2𝜇 − 𝑚 + 1, 𝜏) ]

𝑝

𝑡

ℓ=1

Lemma 6. Let 𝑓(𝑞+2𝑝+𝑚) ∈ 𝐴𝐶[−1, 1] for some 𝑞 ≥ 0, 𝑝, 𝑚 ≥ 1, and

×∑

𝑡!

× ∑ (𝑒2𝑖𝜋(𝑁+ℓ)/(2𝑁+𝑚+1) + 1)

+1 = 𝑂(1/𝑁).

𝑘 = 0, . . . , 𝑞 − 1.

Φ(𝑡) 𝑘,𝑚 (−1)

𝑡=𝑚

(71) We used also the fact that 𝑒

𝑡

𝑡 ∑( ) 𝜇 𝜇=0

× 𝛿±𝑁 (𝜆, (−1)𝑛 𝑒𝑖𝜋𝑛(2𝜇−𝑚+1)/(2𝑁+𝑚+1) )

𝑟 ≠ 0

2𝑖𝜋(𝑁+ℓ)/(2𝑁+𝑚+1)

(𝑗 − 𝑘)!

(−1)

𝑡!

𝑡=0

𝛿𝑛𝑤 (𝛿𝑛𝑝 (𝜆, ∑ 𝐵𝑛+𝑟(2𝑁+𝑚+1) (𝑘))) = 𝑂 (𝑁−2𝑤−2𝑝−𝑘−1 ) .

(1) = 0,

𝐴 𝑘𝑗 (𝑓) (𝑚 + 1)𝑗−𝑘

𝑗−𝑘 (𝑖𝜋)𝑘+1 𝑘=0 2

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

𝑞

1 1 ∑ 2𝑁 + 𝑚 + 1 𝑗=𝑞 𝑁𝑗

×∑

(70)

𝑝

∞

𝑡!

+ 𝑜 (𝑁−𝑞−2𝑝−2 ) .

𝛿±𝑁 (𝜆, 𝐹𝑛,𝑚 ) =

(𝑗 − 𝑘)!(𝑚 + 1)𝑘−𝑗

𝑝

−1 2𝚤𝜋𝑛(𝑠−1)/(2𝑁+𝑚+1) ×∑Vℓ,𝑠 𝑒 ] 𝑠=1

(−1) = 𝑓

𝐴 𝑘𝑗 (𝑓) (2𝑁 + 𝑚 + 1)𝑘

× 𝛿±𝑁 (𝜆, ∑ 𝐵𝑛+𝑟(2𝑁+𝑚+1) (𝑘))

𝑚

𝑓

1 2𝑗 𝑁𝑗

𝑘=0

ℓ=1

(𝑘)

𝑗=𝑞 𝑗

× ∑ (𝑒

(𝑘)

∑

×∑

𝑡

Φ(𝑡) 𝑘,𝑚 (−1)

𝑡=𝑚 𝑚

𝑞+2𝑝+𝑚

𝑝

𝛿±𝑁 (𝜆, 𝐹𝑛,𝑚 ) =

(74)

𝛿−𝑁 (𝜆, 𝐹𝑛,𝑚 ) = (−1)𝑚+1 𝛿𝑁 (𝜆, 𝐹𝑛,𝑚 ) + 𝑂 (𝑁−𝑞−2𝑝−2 ) . (75)

The proof of (77), for 𝑚 = 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


Theorem 7. Let 𝑓(𝑞+2𝑝+𝑚) ∈ 𝐴𝐶[−1, 1] for some 𝑞 ≥ 0, 𝑝, 𝑚 ≥ 1, and 𝑓(𝑘) (−1) = 𝑓(𝑘) (1) = 0,

𝑘 = 0, . . . , 𝑞 − 1.

(78)

Let parameters 𝜆 𝑘 be chosen as in (38). Then, the following estimate holds for |𝑥| < 1: 𝑝

𝑅𝑁,𝑚 (𝑓, 𝑥) =

(−1)𝑁 sin (𝜋𝜎 (𝑁 + (1/2)) 𝑥 − (𝜋𝑚/2)) cos2𝑝+1 (𝜋𝑥/2) (2𝑁)𝑞+2𝑝+1 𝑞

×∑

𝐴 𝑘𝑞 (𝑓) (𝑚 + 1)𝑞−𝑘 2𝑘

𝑘=0

Hence, the last term in the right-hand side of (80) is 𝑜(𝑁−𝑞−2𝑝−1 ). Then, by Lemma 5, 𝛿𝑛1 (𝛿𝑛𝑝 (𝜆, 𝐹𝑛,𝑚 − 𝑓𝑛∗ )) = 𝑂 (𝑁−𝑞−2𝑝−3 ) + 𝑜 (𝑁−𝑞−2𝑝−2 ) , − 𝑁 ≤ 𝑛 ≤ 𝑁 + 𝑚, and the fifth term is also 𝑜(𝑁−𝑞−2𝑝−1 ). Therefore, 𝑝

𝑝

𝑐 (𝑥) 𝑅𝑁,𝑚 (𝑓, 𝑥) = 𝑒−𝑖𝜋𝜎𝑁𝑥 𝛿−𝑁−1 (𝜆, 𝐹𝑛,𝑚 )

(𝑞 − 𝑘)!𝑖𝑘−𝑚 𝜋𝑘+1

𝑝

− 𝑒𝑖𝜋𝜎(𝑁+1)𝑥 𝛿𝑁 (𝜆, 𝐹𝑛,𝑚 )

𝜓+ (𝜏) 𝑚−1 Φ(𝑡) 𝑘,𝑚 (−1) 𝑝 𝑝,𝑚,𝑘 × ((−1) −∑ 𝑘! 𝑡! 𝑡=0

𝑝

+ 𝑒𝑖𝜋𝜎𝑁𝑥 𝛿𝑁+1 (𝜆, 𝐹𝑛,𝑚 )

𝑁 󳨀→ ∞. (79)

Proof. We have from (37) by the Abel transformation (see transformation from (33) to (34) with 𝜆 ≡ 1) 𝑝 −𝑖𝜋𝜎𝑁𝑥 𝛿−𝑁−1

𝑝

(𝜆, 𝐹𝑛,𝑚 ) 𝑐 (𝑥)

𝑅𝑁,𝑚 (𝑓, 𝑥) = 𝑒

𝑝

−𝑒

(𝜆, 𝐹𝑛,𝑚 ) 𝑐 (𝑥)

(80)

𝑝

𝛿−𝑁 (𝜆, 𝐹𝑛,𝑚 ) 𝑐 (𝑥)

+

1 𝑁 1 𝑝 ∑ 𝛿 (𝛿 (𝜆, 𝑓𝑛∗ − 𝐹𝑛,𝑚 )) 𝑒𝑖𝜋𝜎𝑛𝑥 𝑐 (𝑥) 𝑛=−𝑁 𝑛 𝑛

+

1 ∑ 𝛿1 (𝛿𝑝 (𝜆, 𝑓𝑛∗ )) 𝑒𝑖𝜋𝜎𝑛𝑥 , 𝑐 (𝑥) |𝑛|>𝑁 𝑛 𝑛

where 𝑝

𝜋𝜎𝑥 𝑐 (𝑥) = 4cos ( ) ∏ (1 + 𝜆 −𝑠 𝑒𝑖𝜋𝜎𝑥 ) (1 + 𝜆 𝑠 𝑒−𝑖𝜋𝜎𝑥 ) . 2 𝑠=1 (81) 2

It is easy to verify that lim 𝑐 (𝑥) = 22𝑝+2 cos2𝑝+2 (

𝑁→∞

𝜋𝑥 ). 2

(82)

According to Lemma 3, 𝛿𝑛1 (𝛿𝑛𝑝 (𝜆, 𝑓𝑛∗ )) = 𝑂 (𝑛−3 𝑁−2𝑝−𝑞 ) + 𝑜 (𝑛−𝑞−2𝑝−2 ) , |𝑛| > 𝑁,

𝑝

𝑝

𝑐 (𝑥) 𝑅𝑁,𝑚 (𝑓, 𝑥) = 𝛿𝑁 (𝜆, 𝐹𝑛,𝑚 ) × ((−1)𝑚 (𝑒−𝑖𝜋𝜎𝑁𝑥 + 𝑒−𝑖𝜋𝜎(𝑁+1)𝑥 ) (86)

which concludes the proof in view of Lemma 6.

𝑝 𝑖𝜋𝜎𝑁𝑥 𝛿𝑁+1

− 𝑒−𝑖𝜋𝜎(𝑁+1)𝑥

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

+ 𝑜 (𝑁−𝑞−2𝑝−1 )

𝑐 (𝑥)

+𝑒

+ 𝑜 (𝑁−𝑞−2𝑝−1 ) .

− (𝑒𝑖𝜋𝜎𝑁𝑥 + 𝑒𝑖𝜋𝜎(𝑁+1)𝑥 ))

𝑝 𝑖𝜋𝜎(𝑁+1)𝑥 𝛿𝑁 (𝜆, 𝐹𝑛,𝑚 )

(85)

− 𝑒−𝑖𝜋𝜎(𝑁+1)𝑥 𝛿−𝑁 (𝜆, 𝐹𝑛,𝑚 )

𝑡

𝑡 × ∑ ( ) 𝑖2𝜇−𝑚+1 ℎ𝑝 (2𝜇 − 𝑚 + 1, 𝜏)) 𝜇 𝜇=0 + 𝑜 (𝑁−𝑞−2𝑝−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 (−1, 1) and states that for 𝑓(𝑞+2𝑚) ∈ 𝐴𝐶[−1, 1] the convergence rate is 𝑂(𝑁−𝑞−𝑚−1 ) for 𝑚 ≥ 1. Theorem 7 explores the pointwise convergence of the QPR interpolation and shows that convergence rate is 𝑂(𝑁−𝑞−2𝑝−1 ) for 𝑓(𝑞+2𝑝+𝑚) ∈ 𝐴𝐶[−1, 1] and 𝑚 ≥ 1. We see that for 𝑚 = 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 > 𝑚/2 the QPR interpolation has improved accuracy compared to the QP interpolation and improvement is by factor 𝑂(𝑁2𝑝−𝑚 ). 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 𝑚 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 𝑚 small (𝑚 ≤ 6) and additional accuracy obtain by increasing 𝑝.


9 ×10−25

×10−25 8

1.5

6

1

4 0.5

2

−0.6

−0.4

−0.2

0.2

0.4

0.6

−0.6

−0.4

−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 𝜏𝑘 : (−1)𝑝

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

𝑘 = 0, . . . , 𝑞 − 1.

sin (𝜋𝑁𝑥) (−1)𝑁+𝑝 𝑞+2𝑝+1 cos2𝑝+1 (𝜋𝑥/2) (2𝑁) 𝐴 𝑘𝑞 (𝑓) 2𝑘

𝑞

×∑ 𝑘=0

𝑖𝑘 𝜋𝑘+1

+ 𝑜 (𝑁

(𝑞 − 𝑘)!𝑘!

−𝑞−2𝑝−1

+ 𝜓𝑝,0,𝑘 (𝜏)

𝑘!


−∑

𝑡=0

(−1)

𝑡!

𝑡

or + 𝜓𝑝,0,𝑘 (𝜏) = 0,

𝑘 = 0, . . . , 𝑞,

(89)

𝑝

𝑅𝑁,𝑚 (𝑓, 𝑥) = 𝑜 (𝑁−𝑞−2𝑝−1 ) ,

|𝑥| < 1.

(93)

For example, when 𝑝 = 𝑞 = 4 and 𝑚 = 2, we find 𝜏1 = −0.3047909 − 0.4539705𝑖,

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

𝜏2 = 0.9640011 − 0.1481304𝑖, 𝜏3 = 6.375385 + 0.000739𝑖,

4. Parameter Determination in Rational Corrections

(94)

𝜏4 = 12.877189 − 0.000022𝑖.

Till now, we did not discuss the problem of parameters 𝜏𝑘 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 (𝑥) = (1 − 𝑥2 ) sin (𝑥 − 1) .

(91)

𝑘 = 0, . . . , 𝑞

(88)

Let parameters 𝜆 𝑘 be chosen as in (38). Then, the following estimate holds for |𝑥| < 1 as 𝑁 → ∞: 𝑝

+ 𝜓𝑝,𝑚,𝑘 (𝜏)

𝑡 × ∑ ( ) 𝑖2𝜇−𝑚+1 ℎ𝑝 (2𝜇 − 𝑚 + 1, 𝜏) = 0, 𝜇 𝜇=0

Theorem 8. Let 𝑓(𝑞+2𝑝+1) ∈ 𝐴𝐶[−1, 1] for some 𝑞 ≥ 0, 𝑝 ≥ 1, and

𝑅𝑁,0 (𝑓, 𝑥) =

0.6

on interval [−0.7, 0.7] for 𝑚 = 2 and 𝑝 = 4 when 𝜏𝑘 = 𝑘 (a) and with optimal 𝜏𝑘 (see

𝑁 󳨀→ ∞, 𝑞 = 0.

𝑓(𝑘) (−1) = 𝑓(𝑘) (1) = 0,

0.4

(b) 𝑝 |𝑅256,𝑚 (𝑓, 𝑥)|

Note that for 𝑚 = 1 the second term in the brackets of + (1, 𝜏) = 0. Hence, estimate (79) vanishes and also 𝜓𝑝,1,0 𝑅𝑁,1 (𝑓, 𝑥) = 𝑜 (𝑁−2𝑝−1 ) ,

0.2

(90)

One choice is 𝜏𝑘 = 𝑘 which shows satisfactory numerical results (see Figure 1).

4 With this choice 𝑅256,2 (𝑓, 𝑥) = 𝑜(𝑁−13 ) for (90). Figure 1 compares the choices 𝜏𝑘 = 𝑘 (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 𝜆 𝑘 immediately along the ideas of the Fourier-Pade interpolation ([11]). This approach is more complex as 𝜆 𝑘 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 |𝑥| < 1.

10

International Journal of Analysis ×10−35 8

[3] 6

[4]

4 2

[5] 0.7

−0.7

𝑝 Figure 2: The graph of |𝑅256,𝑚 (𝑓, 𝑥)| on interval [−0.7, 0.7] for 𝑚

[6]

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

More specifically, parameters 𝜆 can be determined from the following system: 𝛿𝑛𝑝 (𝜆, 𝐹𝑛,𝑚 ) = 0,

|𝑛| = 𝑁 − 𝑝 + 1, . . . , 𝑁.

(95)

[8]

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

[9]

𝜆 1 = 0.922 + 0.010𝑖,

𝜆 2 = 0.952 + 0.010𝑖,

𝜆 3 = 0.975 + 0.010𝑖,

𝜆 4 = 0.994 − 0.011𝑖,

𝜆 −1 = 0.994 + 0.011𝑖,

𝜆 −2 = 0.975 − 0.010𝑖,

𝜆 −3 = 0.952 − 0.010𝑖,

𝜆 −4 = 0.922 − 0.010𝑖.

[10]

(96)

4 Figure 2 shows the graph of |𝑅256,2 (𝑓, 𝑥)| for function (90) with 𝜆 𝑘 from (96). Comparison with Figure 1 shows high precision of the quasi-periodic Fourier-Pade interpolation for |𝑥| < 1 compared to other choices of parameters 𝜆 𝑘 . 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, “Quasiperiodic interpolation,” Reports of the National Academy of Sciences of Armenia, vol. 101, no. 2, pp. 115–121, 2001. [2] A. Nersessian and N. Hovhannisyan, “Minimization 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, “On a convergence of the quasi-periodic interpolation,” in Proceedings of the 2012 International Workshop on Functional Analysis, Timisoara, Romania, October 2012. L. Poghosyan, “On 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, “Asymptotic estimates for the quasi-periodic interpolations,” Armenian Journal of Mathematics, vol. 5, no. 1, pp. 34–57, 2013. L. Poghosyan and A. Poghosyan, “On a pointwise convergence of the quasi-periodictrigonometric interpolation,” Proceedings of the National Academy of Sciences of Armenia—Mathematics. In press. L. Poghosyan and A. Poghosyan, “Convergence 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, “The 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, “On 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, “On 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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