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Abstract and Applied Analysis
Volume 2018, Article ID 1705409, 21 pages
https://doi.org/10.1155/2018/1705409
Research Article

Optimal Rational Approximations by the Modified Fourier Basis

Institute of Mathematics, Armenian National Academy of Sciences, 24/5 Marshal Baghramyan Ave., 0019 Yerevan, Armenia

Correspondence should be addressed to Arnak V. Poghosyan; ma.ics.htamtsni@kanra

Received 18 October 2017; Accepted 20 February 2018; Published 1 April 2018

Academic Editor: Beong In Yun

Copyright © 2018 Arnak V. Poghosyan and Tigran K. Bakaryan. 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.

Abstract

We consider convergence acceleration of the modified Fourier expansions by rational trigonometric corrections which lead to modified-trigonometric-rational approximations. The rational corrections contain some unknown parameters and determination of their optimal values for improved pointwise convergence is the main goal of this paper. The goal was accomplished by deriving the exact constants of the asymptotic errors of the approximations with further elimination of the corresponding main terms by appropriate selection of those parameters. Numerical experiments outline the convergence improvement of the optimal rational approximations compared to the expansions by the modified Fourier basis.

1. Introduction

The modified Fourier basiswas originally proposed by Krein [1] and thoroughly investigated in a series of papers [210].

Let be the truncated modified Fourier serieswhereObviously, for even functions on , expansions by the modified Fourier basis coincide with the expansions by the classical Fourier basisMoreover, the modified Fourier basis can be derived from the other classical basis on by means of a change of variable.

The first results concerning the convergence of the expansions by the modified Fourier basis appeared in the works [2, 810]. We present two theorems for further comparisons.

Theorem 1 (see [10]). Assume and . If , thenOtherwise,

Under some additional requirements, the convergence rate is faster.

Theorem 2 (see [2, 10]). Assume , , , and obeys the first derivative conditions:If , thenOtherwise,

Overall, we see better convergence rates compared to the classical Fourier expansions [11]. This can be explained by faster decay of coefficients :compared to the classical ones when is smooth enough but nonperiodic on . Estimate (11) can be explained by a nonperiodicity of the basis functions on .

Convergence acceleration of the modified Fourier expansions by means of rational corrections was considered in [6]. Here, we continue those investigations. More specifically, consider a finite sequence of real numbers , and, by , denote the following generalized finite differences:

By , we denote the classical finite differences which correspond to generalized differences with . It is easy to verify that

Letwhere

Consider two sequences of real numbers and . Let and . Let be defined by the following identities:

By means of sequential Abel transformations (see details in [6]), we derive the following expansions of errors (15):whereThese expansions lead to the following modified-trigonometric-rational (MTR-) approximations:with the error

A crucial step for realization of the rational approximations is determination of parameters and . Different approaches are known for solution of this problem (see [1219]). In general, appropriate determination of these parameters leads to rational approximations with improved accuracy compared to the classical ones in case of smooth . However, the rational approximations are essentially nonlinear in the sense thatas for each approximation we need to determine its own and vectors.

In [6], those parameters were determined from the following systems of equations:which led to the Fourier-Pade type approximations [12] with better convergence for smooth functions (see [6]) compared to the expansions by the modified Fourier basis. It is rather complex approach as parameters and depend on and systems (23) must be solved for each .

In this paper, assuming that is smooth on , we consider simpler alternative approach, where and are determined as follows [14, 16, 19]:with and independent of . Actually, in this approach, we take into consideration only the first two terms of the asymptotic expansions of in terms of . Although parameters and in (24) depend on , we need only to determine and which are independent of . Hence, this approach is less complex than the modified Fourier-Pade approximations.

The main results of this paper are exact constants of the main terms of asymptotic errors and optimal parameters for improved pointwise convergence of rational approximations. First, we derive the exact estimates for the main terms of asymptotic errors without specifying parameters and . Second, we determine the optimal values of parameters and which vanish the main terms and lead to approximations with substantially better pointwise convergence rates. We found that optimal values of parameters and , , are the roots of some polynomials depending on and , where indicates the number of zero derivatives in (8). Moreover, the choice of optimal parameters depends on the parity of and also on the location of , whether or .

For example, when is odd and , the roots of the generalized Laguerre polynomial (see Appendix A) could be used for and . In this case, the convergence rate of the MTR-approximations is as . It means better convergence compared to the expansions by the modified Fourier basis with improvement by factor . When is odd and , the roots of the generalized Laguerre polynomial could be used. In this case, the convergence rate is with the improvement by factor . The problem that we encountered is impossibility to get simultaneous optimality on and at . One must decide whether to use parameters that provide the minimal error on , but with worse accuracy on , or work with optimal parameters on with worse accuracy at . We expose similar observations for even values of .

It is important that the values of parameters and depend only on and and are independent of . It means that if functions , , and have enough smoothness and obey the same derivative conditions, the optimal approach leads to linear rational approximations in the sense thatwith the same parameters and for all included functions.

The paper is organized as follows. Section 2 presents some preliminary lemmas. Section 3 explores the pointwise convergence of the MTR-approximations away from the endpoints when . Section 4 considers the pointwise convergence of the MTR-approximations when . In these sections, we show also the results of numerical experiments which confirm and explain the theoretical findings. Section 5 presents some concluding remarks. Appendix A recalls some results concerning the Laguerre polynomials and Appendix B proves some combinatorial identities that we used in the proofs of lemmas and theorems.

2. Preliminaries

Throughout the paper, we assume that parameters , are defined by (see (24))Let and let coefficients be defined by the following identity:

For , , we put

The following estimates can be simply obtained by means of integration by parts of the integrals in (3).

Lemma 3. Assume , , and . Then, the following asymptotic expansions are valid:

Lemma 4. Assume , , and . Then, the following asymptotic expansions are valid:

Next lemma unveils the asymptotic expansions of and , where

Lemma 5. Assume , , , , , andLet , , be defined by (26). Then, the following estimates hold for as :wherewith

Proof. In view of (12) and (26), we haveTaking into account the fact that and using (13), we getApplication of Lemma 3, when is even, and Lemma 4, when + is odd, leads to the following asymptotic expansion ():Then,Finally,It remains to notice that for (see [20]). Hence, , and .

3. Pointwise Convergence Away from the Endpoints

In this section, we explore the pointwise convergence of the MTR-approximations away from the endpoints. Next theorem reveals the asymptotic behavior of the MTR-approximations for without specifying the selection of parameters and .

Theorem 6. Assume , , , , andLet , , be defined by (26). Then, the following estimates hold for as :where

Proof. We prove only estimate (44). Estimate (45) can be handled similarly.
Taking into account the fact that as , we estimate only the sums on the right-hand side of (18). By the Abel transformation, we getLemma 5 estimates sequences , , and as and . It shows that, for and , we haveand the third term in the right-hand side of (47) is . Then, with and , we haveand the second term is . Finally, using the exact estimate for , we derivewhich completes the proof as

Note that Theorem 6 is valid also for which corresponds to the modified Fourier expansions (compare with Theorems 1 and 2). In that case, the exact constants of the main terms in (44) and (45) coincide with similar estimate in [3] (Theorem 2.22, page 29).

Theorem 6 shows thatif parameters and are defined by (24). We see improvement in convergence rate by factor and this result is obtained without specifying parameters and .

Let us compare the modified Fourier expansions and MTR-approximations for a specific smooth function. Consider the following one:for whichHence, this function obeys first derivative conditions (8).

Figures 1 and 2 show the behaviors of () and (, , , ), respectively, on interval for while approximating (53). We used , in the rational approximations.

Figure 1: The graph of on for while approximating (53) by the modified Fourier expansion (2).
Figure 2: The graphs of on for and different while approximating (53) by the MTR-approximations with , .

According to the results of Theorem 6, the bigger the value of is, the higher the accuracy of the corresponding approximations is. We observe it empirically. We see that is for , is for , is for , is for , and is for .

Can we improve the accuracy of the rational approximations by appropriate selection of parameters and ? Further, in this section, we give positive answer to this question and show how the optimal values can be chosen.

Estimates of Theorem 6 show that improvement can be achieved if parameters are chosen such that andBy looking into the definition of , we observe that condition (55) can be achieved, for example, ifwhere is a polynomial of order with unknown coefficients , . Then, condition (55) follows from the well-known identityFurther, we determine the values of , , for improved convergence of the rational approximations.

Next result is an immediate consequence of those observations and estimates of Theorem 6.

Theorem 7. Let , , , , andAssume the polynomialhas only real-valued and nonzero roots , , and letThen, the following estimate holds for :

Theorem 7 is valid only if, for given and , polynomial (60) has only real-valued and nonzero roots. Further, we clarify this statement by showing those cases when it is true.

By imposing extra smoothness on the underlying functions, we derive more precise estimate of (62).

First, we need estimates for and .

Lemma 8. Assume , , , , andAssume polynomial (60) has only real-valued and nonzero roots , , and is defined by (26) with . Let , when and have the same parity, and , otherwise. Then, the following estimates hold as :wherewith and defined in Lemma 5.

Proof. We prove only (64). By taking in (33) and using (56), we getThis completes the proof in view of (B.15) and (B.16).

Further, in Theorems 9 and 10, we show that the pointwise convergence rate of the rational approximations depends on the asymptotic and . From the other side, Lemma 8 reveals that the convergence rates of those sequences depend on the value of (as )When is odd, for the highest power of , parameter can be only . It means that . When is even, parameter can be which means that . We determine parameter later.

Next theorem unveils the convergence rate of the MTR-approximations for odd values of , when . Note that, in this case, the roots of polynomial (60) coincide with the roots of generalized Laguerre polynomial (see Appendix A).

Theorem 9. Let parameter be odd, , , , andLet , , be defined by (26), where are the roots of the generalized Laguerre polynomial . Then, the following estimates hold for :where and are defined in Lemma 8.

Proof. We use (18) to estimate . The error can be estimated similarly.
Taking into account the fact that as , we estimate only the sums on the right-hand side of (18). An application of the Abel transformation to the sums of leads to the following expansion of the error:According to Lemma 5, we haveand hence the last term on the right-hand side of (71) is as . It follows from Lemma 8 that the third term in (71) is as . Therefore,By Lemma 8, we haveTaking into account the fact that (see (B.16))we conclude that and are nonzero only for which leads to the following estimates:From here, we getwhich completes the proof.

Figure 3 shows the result of approximation of (53) by the MTR-approximations with optimal parameters , as the roots of . We see better accuracy on compared to nonoptimal parameters as in Figure 2. For , the improvement is almost times, and for , the improvement is almost 240 times.