On a Fast Convergence of the Rational-Trigonometric-Polynomial Interpolation

Abstract

We consider the convergence acceleration of the Krylov-Lanczos interpolation by rational correction functions and investigate convergence of the resultant parametric rational-trigonometric-polynomial interpolation. Exact constants of asymptotic errors are obtained in the regions away from discontinuities, and fast convergence of the rational-trigonometric-polynomial interpolation compared to the Krylov-Lanczos interpolation is observed. Results of numerical experiments confirm theoretical estimates and show how the parameters of the interpolations can be determined in practice.

1. Introduction

In this paper, we continue investigations started in [1] where we considered the convergence acceleration of the classical trigonometric interpolation 𝐼_𝑁𝑓,𝑥=^𝑁̌_𝑛=−𝑁𝑓_𝑛𝑒^{𝑖𝜋𝑛𝑥},_̌‌𝑓_𝑛=12𝑁+1^𝑁_𝑘=−𝑁𝑓𝑥_𝑘𝑒^{−𝑖𝜋𝑛𝑥𝑘},𝑥_𝑘=2𝑘,2𝑁+1 (1)(2) via polynomial corrections representing discontinuities of the function 𝑓 and some of its first 𝑞 derivatives (jumps). The resultant interpolation 𝐼_𝑁,𝑞(𝑓,𝑥) was called as the Krylov-Lanczos (KL-) interpolation. That approach was suggested in 1906 by Krylov [2] and later in 1964 by Lanczos [3, 4] (see also [1, 5–10] with references therein).

Here, we consider the convergence acceleration of the KL-interpolation by the application of rational (by 𝑒^𝑖𝜋𝑥) correction functions along the ideas of the rational approximations (see [11–13] with references therein). The approach discussed here leads to the parametric (depending on parameters 𝜃₁,…,𝜃_𝑝) rational-trigonometric-polynomial (rtp-) interpolation 𝐼^𝑝_𝑁,𝑞(𝑓,𝑥). The idea of the convergence acceleration via sequential application of polynomial and rational corrections was described in [14–17]. The KL-interpolation is a special case of the rtp-interpolation corresponding to the choice of parameters 𝜃_𝑘=0, 𝑘=1,…,𝑝. Besides, rational corrections can be applied immediately to the classical interpolation without polynomial corrections (see interpolation 𝐼^𝑝_𝑁(𝑓,𝑥)).

In this paper, we reveal the convergence properties of the rtp-interpolation, show its fast convergence compared to the KL-interpolation in the regions away from the singularities (𝑥=±1), and discuss the problem of parameters determination in rational corrections.

2. Rational Interpolations

In this section, we introduce a rational interpolation as a method of the convergence acceleration of the classical trigonometric interpolation. Here, we recap some details from [15, 16].

By 𝑟_𝑁(𝑓,𝑥), we denote the error of the classical trigonometric interpolation 𝑟_𝑁𝑓,𝑥=𝑓(𝑥)−𝐼_𝑁𝑓,𝑥, (3) and write 𝑟_𝑁𝑓,𝑥=^𝑁_𝑛=−𝑁𝑓_𝑛−𝑓^̌‌_𝑛𝑒^{𝑖𝜋𝑛𝑥}+^∞_𝑛=𝑁+1𝑓_𝑛𝑒^{𝑖𝜋𝑛𝑥}+^−𝑁−1_𝑛=−∞𝑓_𝑛𝑒^{𝑖𝜋𝑛𝑥}, (4) where 𝑓_𝑛 is the 𝑛th Fourier coefficient of 𝑓𝑓_𝑛=12¹₋₁𝑓(𝑥)𝑒^{−𝑖𝜋𝑛𝑥}𝑑𝑥. (5)

Rational corrections considered in this paper are based on a series of formulae of summation by parts applied to the error terms in (4). Such transformations lead to new interpolations with correction terms in the form of rational (by 𝑒^𝑖𝜋𝑥) functions. Consider a vector of complex numbers 𝜃={𝜃₁,…,𝜃_𝑝}. The first formula of summation by parts is easy to verify straightforwardly as ^∞_𝑛=𝑁+1𝑓_𝑛𝑒^{𝑖𝜋𝑛𝑥}=−𝜃₁𝑓_𝑁𝑒^{𝑖𝜋(𝑁+1)𝑥}1+𝜃₁𝑒^𝑖𝜋𝑥+11+𝜃₁𝑒^{∞𝑖𝜋𝑥}_𝑛=𝑁+1𝑓_𝑛+𝜃₁𝑓_𝑛−1𝑒^{𝑖𝜋𝑛𝑥}. (6) If here |𝜃₁|<1, then the formula is valid for all |𝑥|≤1. This is the main reason of including the parameters 𝜃_𝑘 here and further. If 𝜃₁ is chosen appropriately (see (32)), then the second term in the right-hand side of (6) converges faster (however, for |𝑥|<1) than the sum in the left-hand side. The next, slightly different formula of summation by parts is also easy to derive as ^∞_𝑛=𝑁+1𝑓_𝑛𝑒^{𝑖𝜋𝑛𝑥}=𝜃₁𝑓_𝑁+1𝑒^{𝑖𝜋𝑁𝑥}1+𝜃₁𝑒^{−𝑖𝜋𝑥}+11+𝜃₁𝑒^{∞−𝑖𝜋𝑥}_𝑛=𝑁+1𝑓_𝑛+𝜃₁𝑓_𝑛+1𝑒^{𝑖𝜋𝑛𝑥}. (7) Application of (7) to the second term in the right-hand side of (6) leads to the needed expansion ^∞_𝑛=𝑁+1𝑓_𝑛𝑒^{𝑖𝜋𝑛𝑥}=11+𝜃₁𝑒^𝑖𝜋𝑥1+𝜃₁𝑒^{−𝑖𝜋𝑥}×^∞_𝑛=𝑁+1𝑓_𝑛+𝜃₁𝑓_𝑛−1+𝜃₁𝑓_𝑛+1+𝜃₁𝑓_𝑛𝑒^{𝑖𝜋𝑛𝑥}−𝜃₁𝑓_𝑁𝑒^{𝑖𝜋(𝑁+1)𝑥}1+𝜃₁𝑒^𝑖𝜋𝑥1+𝜃₁𝑒^{−𝑖𝜋𝑥}+𝜃₁𝑓_𝑁+1𝑒^{𝑖𝜋𝑁𝑥}1+𝜃₁𝑒^𝑖𝜋𝑥1+𝜃₁𝑒^{−𝑖𝜋𝑥}. (8) Here, also, as we mentioned above, the sum on the right-hand side converges faster than on the left-hand side if parameter 𝜃₁ is chosen appropriately.

In a similar manner, we transform the third term in the right-hand side of (4): ^−𝑁−1_𝑛=−∞𝑓_𝑛𝑒^{𝑖𝜋𝑛𝑥}=11+𝜃₁𝑒^𝑖𝜋𝑥1+𝜃₁𝑒^{−𝑖𝜋𝑥}×^−𝑁−1_𝑛=−∞𝑓_𝑛+𝜃₁𝑓_𝑛−1+𝜃₁𝑓_𝑛+1+𝜃₁𝑓_𝑛𝑒^{𝑖𝜋𝑛𝑥}−𝜃₁𝑓_−𝑁𝑒^{−𝑖𝜋(𝑁+1)𝑥}1+𝜃₁𝑒^𝑖𝜋𝑥1+𝜃₁𝑒^{−𝑖𝜋𝑥}+𝜃₁𝑓_−𝑁−1𝑒^{−𝑖𝜋𝑁𝑥}1+𝜃₁𝑒^𝑖𝜋𝑥1+𝜃₁𝑒^{−𝑖𝜋𝑥}. (9)

For the first term in the right-hand side of (4), the formula of summation by parts is the following: ^𝑁_𝑛=−𝑁𝑐_𝑛𝑒^{𝑖𝜋𝑛𝑥}=11+𝜃₁𝑒^𝑖𝜋𝑥1+𝜃₁𝑒^{−𝑖𝜋𝑥}×^𝑁_𝑛=−𝑁𝑐_𝑛+𝜃₁𝑐_𝑛−1+𝜃₁𝑐_𝑛+1+𝜃₁𝑐_𝑛𝑒^{𝑖𝜋𝑛𝑥}+𝜃₁𝑐_𝑁𝑒^{𝑖𝜋(𝑁+1)𝑥}1+𝜃₁𝑒^𝑖𝜋𝑥1+𝜃₁𝑒^{−𝑖𝜋𝑥}−𝜃₁𝑐_−𝑁−1𝑒^{−𝑖𝜋𝑁𝑥}1+𝜃₁𝑒^𝑖𝜋𝑥1+𝜃₁𝑒^{−𝑖𝜋𝑥}−𝜃₁𝑐_𝑁+1𝑒^{𝑖𝜋𝑁𝑥}1+𝜃₁𝑒^𝑖𝜋𝑥1+𝜃₁𝑒^{−𝑖𝜋𝑥}+𝜃₁𝑐_−𝑁𝑒^{−𝑖𝜋(𝑁+1)𝑥}1+𝜃₁𝑒^𝑖𝜋𝑥1+𝜃₁𝑒^{−𝑖𝜋𝑥}, (10) where 𝑐_𝑛=𝑓_𝑛−𝑓^̌‌_𝑛.

Substituting (8), (9), and (10) into (4), after some simplifications, provides with the following expansion of the error 𝑟_𝑁𝑓,𝑥=𝜃𝑓₁^̌_𝑁𝑒^{−𝑖𝜋𝑁𝑥}−𝑒^{𝑖𝜋(𝑁+1)𝑥}1+𝜃₁𝑒^𝑖𝜋𝑥1+𝜃₁𝑒^{−𝑖𝜋𝑥}+𝜃𝑓₁^̌_−𝑁𝑒^{𝑖𝜋𝑁𝑥}−𝑒^{−𝑖𝜋(𝑁+1)𝑥}1+𝜃₁𝑒^𝑖𝜋𝑥1+𝜃₁𝑒^{−𝑖𝜋𝑥}+11+𝜃₁𝑒^𝑖𝜋𝑥1+𝜃₁𝑒^{−𝑖𝜋𝑥}×^∞_{|𝑛|=𝑁+1}𝑓_𝑛+𝜃₁𝑓_𝑛−1+𝜃₁𝑓_𝑛+1+𝜃₁𝑓_𝑛𝑒^{𝑖𝜋𝑛𝑥}+11+𝜃₁𝑒^𝑖𝜋𝑥1+𝜃₁𝑒^{−𝑖𝜋𝑥}×^𝑁_𝑛=−𝑁𝑐_𝑛+𝜃₁𝑐_𝑛−1+𝜃₁𝑐_𝑛+1+𝜃₁𝑐_𝑛𝑒^{𝑖𝜋𝑛𝑥}, (11) where 𝑐_𝑛=𝑓_𝑛−𝑓^̌‌_𝑛. Here, we also took into account the periodicity of the coefficients 𝑓^̌‌_𝑛𝑓^̌‌_𝑁+1=𝑓^̌‌_−𝑁,𝑓^̌‌_−𝑁−1=𝑓^̌‌_𝑁. (12) For writing the expansion (11) in a short form and also for further reiterations of this transformation, we introduce the following generalized finite differences 𝛿^𝑘_𝑛(𝜃,𝑦_𝑛) determined recurrently: 𝛿⁰_𝑛𝜃,𝑦_𝑛=𝑦_𝑛,𝛿^𝑘_𝑛𝜃,𝑦_𝑛=𝛿_𝑛^𝑘−1𝜃,𝑦_𝑛+𝜃_𝑘𝛿^𝑘−1_𝑛−1𝜃,𝑦_𝑛+𝜃_𝑘𝛿^𝑘−1_𝑛+1𝜃,𝑦_𝑛+𝜃_𝑘𝛿_𝑛^𝑘−1𝜃,𝑦_𝑛, (13) for some sequence 𝑦_𝑛. Now, (11) can be rewritten in the form 𝑟_𝑁𝑓,𝑥=𝜃₁𝛿⁰_𝑁𝜃,𝑓^̌‌_𝑛𝑒^{−𝑖𝜋𝑁𝑥}−𝑒^{𝑖𝜋(𝑁+1)𝑥}1+𝜃₁𝑒^𝑖𝜋𝑥1+𝜃₁𝑒^{−𝑖𝜋𝑥}+𝜃₁𝛿⁰_−𝑁𝜃,𝑓^̌‌_𝑛𝑒^{𝑖𝜋𝑁𝑥}−𝑒^{−𝑖𝜋(𝑁+1)𝑥}1+𝜃₁𝑒^𝑖𝜋𝑥1+𝜃₁𝑒^{−𝑖𝜋𝑥}+11+𝜃₁𝑒^𝑖𝜋𝑥1+𝜃₁𝑒^{−𝑖𝜋𝑥}×^∞_{|𝑛|=𝑁+1}𝛿¹_𝑛𝜃,𝑓_𝑛𝑒^{𝑖𝜋𝑛𝑥}+11+𝜃₁𝑒^𝑖𝜋𝑥1+𝜃₁𝑒^{−𝑖𝜋𝑥}×^𝑁_𝑛=−𝑁𝛿¹_𝑛𝜃,𝑓_𝑛−𝑓^̌‌_𝑛𝑒^{𝑖𝜋𝑛𝑥}. (14) Reiteration of this transformation up to 𝑝 times leads to the following expansion of the error: 𝑟_𝑁=𝑒𝑓,𝑥_{−𝑖𝜋𝑁𝑥}−𝑒^{𝑖𝜋(𝑁+1)𝑥}×^𝑝_𝑘=1𝜃_𝑘𝛿_𝑁^𝑘−1𝜃,𝑓^̌‌_𝑛∏^𝑘_𝑠=11+𝜃_𝑠𝑒^𝑖𝜋𝑥1+𝜃_𝑠𝑒^{−𝑖𝜋𝑥}+𝑒^{𝑖𝜋𝑁𝑥}−𝑒^{−𝑖𝜋(𝑁+1)𝑥}×^𝑝_𝑘=1𝜃_𝑘𝛿^𝑘−1_−𝑁𝜃,𝑓^̌‌_𝑛∏^𝑘_𝑠=11+𝜃_𝑠𝑒^𝑖𝜋𝑥1+𝜃_𝑠𝑒^{−𝑖𝜋𝑥}+1∏^𝑝_𝑠=11+𝜃_𝑠𝑒^𝑖𝜋𝑥1+𝜃_𝑠𝑒^{−𝑖𝜋𝑥∞}_{|𝑛|=𝑁+1}𝛿^𝑝_𝑛𝜃,𝑓_𝑛𝑒^{𝑖𝜋𝑛𝑥}+1∏^𝑝_𝑠=11+𝜃_𝑠𝑒^𝑖𝜋𝑥1+𝜃_𝑠𝑒^{−𝑖𝜋𝑥}×^𝑁_𝑛=−𝑁𝛿^𝑝_𝑛𝜃,𝑓_𝑛−𝑓^̌‌_𝑛𝑒^{𝑖𝜋𝑛𝑥}, (15) where the first two terms in the right-hand side can be viewed as rational corrections to the error and the last two terms as the actual error. This observation leads to the following rational-trigonometric interpolation: 𝐼^𝑝_𝑁𝑓,𝑥=𝐼_𝑁𝑓,𝑥+𝑒^{−𝑖𝜋𝑁𝑥}−𝑒^{𝑖𝜋(𝑁+1)𝑥}×^𝑝_𝑘=1𝜃_𝑘𝛿_𝑁^𝑘−1𝜃,𝑓^̌‌_𝑛∏^𝑘_𝑠=11+𝜃_𝑠𝑒^𝑖𝜋𝑥1+𝜃_𝑠𝑒^{−𝑖𝜋𝑥}+𝑒^{𝑖𝜋𝑁𝑥}−𝑒^{−𝑖𝜋(𝑁+1)𝑥}×^𝑝_𝑘=1𝜃_𝑘𝛿^𝑘−1_−𝑁𝜃,𝑓^̌‌_𝑛∏^𝑘_𝑠=11+𝜃_𝑠𝑒^𝑖𝜋𝑥1+𝜃_𝑠𝑒^{−𝑖𝜋𝑥}, (16) with the error 𝑟^𝑝_𝑁=1𝑓,𝑥∏^𝑝_𝑠=11+𝜃_𝑠𝑒^𝑖𝜋𝑥1+𝜃_𝑠𝑒^{−𝑖𝜋𝑥}×^∞_{|𝑛|=𝑁+1}𝛿^𝑝_𝑛𝜃,𝑓_𝑛𝑒^{𝑖𝜋𝑛𝑥}+1∏^𝑝_𝑠=11+𝜃_𝑠𝑒^𝑖𝜋𝑥1+𝜃_𝑠𝑒^{−𝑖𝜋𝑥}×^𝑁_𝑛=−𝑁𝛿^𝑝_𝑛𝜃,𝑓_𝑛−𝑓^̌‌_𝑛𝑒^{𝑖𝜋𝑛𝑥}. (17)

The problem of the determination of parameters 𝜃_𝑘 will be discussed later.

3. The KL-Interpolation

In this section, we consider the additional acceleration of the rational-trigonometric interpolation by the polynomial correction method known as the Krylov-Lanczos approach. We recap the main ideas from [1].

Let 𝑓∈𝐶^𝑞−1[−1,1]. By 𝐴_𝑘(𝑓), we denote the jumps of 𝑓 at the end points of the interval 𝐴_𝑘𝑓=𝑓^(𝑘)(1)−𝑓^(𝑘)(−1),𝑘=0,…,𝑞−1. (18)

The polynomial correction method is based on the following representation of the interpolated function: 𝑓(𝑥)=^𝑞−1_𝑘=0𝐴_𝑘𝑓𝐵_𝑘(𝑥)+𝐹(𝑥), (19) where 𝐵_𝑘 are 2-periodic Bernoulli polynomials 𝐵₀(𝑥)=𝑥2,𝐵_𝑘(𝑥)=𝐵_𝑘−1(𝑥)𝑑𝑥,₁₋₁𝐵_𝑘(𝑥)𝑑𝑥=0,𝑥∈[−1,1], (20) with the Fourier coefficients 𝐵_𝑛(𝑘)=(−1)^𝑛+12(𝑖𝜋𝑛)^𝑘+1,𝑛≠0,𝐵₀(𝑘)=0. (21) Function 𝐹 is a 2-periodic and relatively smooth function on the real line (𝐹∈𝐶^𝑞−1(𝑅)) with the discrete Fourier coefficients 𝐹^̌‌_𝑛=𝑓^̌‌_𝑛−^𝑞−1_𝑘=0𝐴_𝑘𝑓𝐵^̌‌_𝑛(𝑘). (22) The approximation of 𝐹 in (19) by the classical trigonometric interpolation leads to the Krylov-Lanczos (KL-) interpolation 𝐼_𝑁,𝑞𝑓,𝑥=^𝑞−1_𝑘=0𝐴_𝑘𝑓𝐵_𝑘(𝑥)+𝐼_𝑁(𝐹,𝑥), (23) and the approximation of 𝐹 by the rational-trigonometric interpolation leads to the rational-trigonometric-polynomial (rtp-) interpolation 𝐼^𝑝_𝑁,𝑞𝑓,𝑥=^𝑞−1_𝑘=0𝐴_𝑘𝑓𝐵_𝑘(𝑥)+𝐼^𝑝_𝑁(𝐹,𝑥) (24) with the errors 𝑟_𝑁,𝑞𝑓,𝑥=𝑓(𝑥)−𝐼_𝑁,𝑞,𝑟𝑓,𝑥_{𝑝𝑁,𝑞}𝑓,𝑥=𝑓(𝑥)−𝐼^𝑝_𝑁,𝑞𝑓,𝑥, (25)(26) respectively.

We need the next results for further comparisons. Theorems 1 and 2 show the behavior of the KL-interpolation in the regions away from the singularities (𝑥=±1).

Denote 𝜙_𝑚=^∞_𝑠=−∞(−1)^𝑠(2𝑠+1)^𝑚. (27)

Theorem 1 (see [1]). Let 𝑞≥2 be even, 𝑓∈𝐶^𝑞+1[−1,1], and 𝑓^(𝑞+1)∈𝐴𝐶[−1,1]. Then, the following estimate holds for |𝑥|<1, 𝑟_𝑁,𝑞=𝜑𝑓,𝑥^𝑁,𝑞𝑓,𝑥𝑁^𝑞+1+𝑜𝑁^−𝑞−1,𝑁→∞, (28) where 𝜑_𝑁,𝑞𝑓,𝑥=𝐴_𝑞𝑓(−1)^𝑁+𝑞/22𝜋^𝑞+1sin(𝜋(2𝑁+1)𝑥/2)()𝜙cos𝜋𝑥/2^𝑞+1. (29)

Theorem 2 (see [1]). Let 𝑞≥1 be odd, 𝑓∈𝐶^𝑞+2[−1,1], and 𝑓^(𝑞+2)∈𝐴𝐶[−1,1]. Then, the following estimate holds for |𝑥|<1, 𝑟_𝑁,𝑞=𝜑𝑓,𝑥^𝑁,𝑞𝑓,𝑥𝑁^𝑞+2+𝑜𝑁^−𝑞−2,𝑁→∞, (30) where 𝜑_𝑁,𝑞𝑓,𝑥=𝐴_𝑞𝑓(−1)^{𝑁+(𝑞+1)/2+1}𝑞+14𝜋^𝑞+1×sin(𝜋𝑥/2)sin(𝜋(2𝑁+1)𝑥/2)cos²()𝜙𝜋𝑥/2^𝑞+2+𝐴_𝑞+1𝑓(−1)^{𝑁+(𝑞+1)/2}2𝜋^𝑞+2sin(𝜋(2𝑁+1)𝑥/2)()𝜙cos𝜋𝑥/2^𝑞+2. (31)

The aim of this paper is the derivation of analogs of Theorems 1 and 2 for the rtp-interpolations.

The determination of parameters 𝜃_𝑘 is crucial for the realization of the rtp-interpolation. General method leads to the Fourier-Padé interpolation (see [16]).

Here, we consider a smooth function 𝑓 on [−1,1] and take 𝜃_𝑘𝜏=1−^𝑘𝑁,𝑘=1,…,𝑝, (32) where the new parameters 𝜏_𝑘 are independent of 𝑁. They can be determined differently. One approach leads to the 𝐿₂-minimal interpolation [14, 18]. This idea was introduced and investigated in [14] for the Fourier-Padé approximations. The first step towards 𝐿₂-minimal interpolation was performed in [18]. The idea of this interpolation was in the determination of unknown parameters 𝜏_𝑘 from the condition lim_𝑁→∞𝑁^𝑞+1/2𝑟^𝑝_𝑁,𝑞𝑓,𝑥_𝐿2→minimum. (33) Paper [18] showed the solution of that problem for 𝑝=1 and 1≤𝑞≤6.

Another approach for the determination of parameters 𝜏_𝑘 was described in [17], where 𝜏_𝑘 were the roots of the associated Laguerre polynomial 𝐿^𝑞_𝑝(𝑥). Below, in numerical experiments, we use this approach.

The following theorem, proved in [15], shows the convergence rate of the rtp-interpolation in the regions away from the singularities 𝑥=±1 for parameters 𝜃_𝑘 chosen as in (32).

Theorem 3 (see [15]). Let 𝑓∈𝐶^𝑞+2𝑝+1[−1,1] and 𝑓^{(𝑞+2𝑝+1)}∈𝐴𝐶[−1,1] for some 𝑞,𝑝≥1. Let parameters 𝜃_𝑘 be chosen as in (32). Then, the following estimate holds as 𝑁→∞ and |𝑥|<1𝑟^𝑝_𝑁,𝑞𝑓,𝑥=𝑂𝑁^{−𝑞−2𝑝−1}. (34)

The main purpose of this paper is the derivation of the exact estimates of the 𝑟^𝑝_𝑁,𝑞(𝑓,𝑥) when |𝑥|<1 getting more accurate ones than Theorem 3 presents. Based on such estimates, we discuss the problem of parameters (𝜃,𝑝, and 𝑞) determination for accurate interpolation. Theoretical estimates show the fast convergence of the rtp-interpolation compared to the KL-interpolation. Numerical experiments confirm theoretical estimates.

Throughout the paper, it is supposed that the exact values of the jumps 𝐴_𝑘(𝑓) are known and that interpolated function is smooth on [−1,1].

4. Pointwise Convergence of the RTP-Interpolation

Let 𝜃_𝑘 be defined as in (32) and by 𝛾_𝑘(𝜏) denote the coefficients of the polynomial ^𝑝_𝑠=11+𝜏_𝑠𝑥=^𝑝_𝑠=0𝛾_𝑠(𝜏)𝑥^𝑠. (35)

Also denote 𝜓_𝑚,𝑝=^𝑝_𝑠=0(−1)^𝑠𝛾_𝑠(𝜏)^𝑝_𝑘=0𝛾_𝑘(𝜏)2𝑝−𝑘−𝑠+𝑚!𝜙_{2𝑝−𝑘−𝑠+𝑚+1}. (36)

Theorems 4 and 5 describe pointwise behavior of the rtp-interpolation in the regions away from the singularities 𝑥=±1. Theorem 4 deals with even values of 𝑞 and Theorem 5 with odd values.

Theorem 4. Let 𝑞≥2 be even and 𝑓∈𝐶^𝑞+2𝑝+1[−1,1] with 𝑓^{(𝑞+2𝑝+1)}∈𝐴𝐶[−1,1] for some 𝑝≥1. Let parameters 𝜃_𝑘 be chosen as in (32). Then, the following estimate holds for |𝑥|<1, 𝑟^𝑝_𝑁,𝑞=𝜑𝑓,𝑥^{𝑁,𝑞,𝑝}𝑓,𝑥𝑁^2𝑝+𝑞+1+𝑜𝑁^{−2𝑝−𝑞−1},𝑁→∞, (37) where 𝜑_{𝑁,𝑞,𝑝}𝑓,𝑥=𝐴_𝑞𝑓(−1)^{𝑁+𝑝+𝑞/2}2^2𝑝+1𝜋^𝑞+1(𝜋())𝑞!sin2𝑁+1𝑥/2cos^2𝑝+1()𝜓𝜋𝑥/2^𝑞,𝑝. (38)

Proof. Expansion (19) and definition of interpolation 𝐼^𝑝_𝑁,𝑞(𝑓,𝑥) show that 𝑟^𝑝_𝑁,𝑞𝑓,𝑥=𝑟^𝑝_𝑁(𝐹,𝑥). (39) The application of transformation (14) to (17) with 𝜃_𝑝+1=1 implies 𝑟^𝑝_𝑁,𝑞=𝛿𝑓,𝑥^𝑝𝑁𝜃,𝐹^̌‌_𝑛𝑐(𝑥)𝑒^{−𝑖𝜋𝑁𝑥}−𝑒^{𝑖𝜋(𝑁+1)𝑥}+𝛿^𝑝_−𝑁𝜃,𝐹^̌‌_𝑛𝑐(𝑥)𝑒^{𝑖𝜋𝑁𝑥}−𝑒^{−𝑖𝜋(𝑁+1)𝑥}+1𝑐(𝑥)^𝑁_𝑛=−𝑁𝛿¹_𝑛𝛿^𝑝_𝑛𝜃,𝐹_𝑛−𝐹^̌‌_𝑛𝑒^{𝑖𝜋𝑛𝑥}+1𝑐(𝑥)^∞_{|𝑛|=𝑁+1}𝛿¹_𝑛𝛿^𝑝_𝑛𝜃,𝐹_𝑛𝑒^{𝑖𝜋𝑛𝑥}, (40) where, by 𝛿^𝑠_𝑛(𝑦_𝑛), we denoted 𝛿^𝑠_𝑛(𝜃,𝑦_𝑛) with 𝜃_𝑘=1, 𝑘=1,…,𝑠 and 𝑐(𝑥)=1+𝑒^𝑖𝜋𝑥1+𝑒^{−𝑖𝜋𝑥𝑝}_𝑠=11+𝜃_𝑠𝑒^𝑖𝜋𝑥1+𝜃_𝑠𝑒^{−𝑖𝜋𝑥}=4cos²𝜋𝑥2^𝑝_𝑠=11+2𝜃_𝑠cos𝜋𝑥+𝜃²_𝑠. (41)
First, we estimate the last term in the right-hand side of (40). We need to estimate 𝛿¹_𝑛(𝛿^𝑝_𝑛(𝜃,𝐹_𝑛)) for |𝑛|>𝑁 as 𝑁→∞. In view of the smoothness of 𝐹, we get from expansion (19) by means of integration by parts 𝐹_𝑛=^𝑞+2𝑝+1_𝑚=𝑞𝐴_𝑚𝑓𝐵_𝑛(𝑚)+12(𝑖𝜋𝑛)^2𝑝+𝑞+2×¹₋₁𝑓^{(2𝑝+𝑞+2)}(𝑥)𝑒^{−𝑖𝜋𝑛𝑥}=𝑑𝑥_{𝑞+2𝑝+1𝑚=𝑞}𝐴_𝑚𝑓𝐵_𝑛(𝑚)+𝑜𝑛^{−2𝑝−𝑞−2}, (42) and consequently 𝛿¹_𝑛𝛿^𝑝_𝑛𝜃,𝐹_𝑛=^𝑞+2𝑝+1_𝑚=𝑞𝐴_𝑚𝑓𝛿¹_𝑛𝛿^𝑝_𝑛𝜃,𝐵_𝑛(𝑚)+𝑜𝑛^{−2𝑝−𝑞−2}. (43) According to Lemma A.1, 𝛿¹_𝑛𝛿^𝑝_𝑛𝜃,𝐵_𝑛(𝑚=1)𝑁^2𝑝𝑂𝑛^−𝑚−3, (44) and, therefore, 𝛿¹_𝑛𝛿^𝑝_𝑛𝜃,𝐹_𝑛=1𝑁^2𝑝𝑜𝑛^−𝑞−2. (45) Thus, we conclude that the last term in the right-hand side of (40) is 𝑜(𝑁^{−𝑞−2𝑝−1}).
Now, we estimate the third term in the right-hand side of (40). We need to estimate 𝛿¹_𝑛(𝛿^𝑝_𝑛(𝜃,𝐹_𝑛−𝐹^̌‌_𝑛)) for |𝑛|≤𝑁 as 𝑁→∞. Observing that 𝐹^̌‌_𝑛−𝐹_𝑛=_𝑠≠0𝐹_{𝑛+𝑠(2𝑁+1)}, (46) and, applying (42), we obtain 𝐹^̌‌_𝑛−𝐹_𝑛=^𝑞+2𝑝+1_𝑚=𝑞𝐴_𝑚𝑓_𝑠≠0𝐵_{𝑛+𝑠(2𝑁+1)}+1𝑁^2𝑝+𝑞+2×_𝑠≠0𝜀_{𝑛+𝑠(2𝑁+1)}(𝑛/(2𝑁+1)+𝑠)^𝑞+2𝑝+2=^𝑞+2𝑝+1_𝑚=𝑞𝐴_𝑚𝑓𝐵^̌‌_𝑛(𝑚)−𝐵_𝑛(𝑚)+𝑜𝑁^{−2𝑝−𝑞−2},|𝑛|≤𝑁,𝑁→∞, (47) where, by 𝜀_𝑛, we denoted 𝜀_𝑛=12(𝑖𝜋)^{2𝑝+𝑞+21−1}𝑓^{(2𝑝+𝑞+2)}(𝑥)𝑒^{−𝑖𝜋𝑛𝑥}𝑑𝑥 (48) and took into account that 𝜀_𝑛=𝑜(1) as 𝑛→∞. Now, it follows that 𝛿¹_𝑛𝛿^𝑝_𝑛𝜃,𝐹^̌‌_𝑛−𝐹_𝑛=^𝑞+2𝑝+1_𝑚=𝑞𝐴_𝑚𝑓𝛿¹_𝑛𝛿^𝑝_𝑛𝜃,𝐵^̌‌_𝑛(𝑚)−𝐵_𝑛(𝑚)+𝑜𝑁^{−2𝑝−𝑞−2}. (49) According to Lemma A.2, 𝛿¹_𝑛𝛿^𝑝_𝑛𝜃,𝐵^̌‌_𝑛(𝑚)−𝐵_𝑛(𝑚)=𝑂𝑁^{−2𝑝−𝑞−3}, (50) and, therefore, 𝛿¹_𝑛𝛿^𝑝_𝑛𝜃,𝐹^̌‌_𝑛−𝐹_𝑛=𝑜𝑁^{−2𝑝−𝑞−2}. (51) Hence, the third term in the right-hand side of (40) is also 𝑜(𝑁^{−𝑞−2𝑝−1}) as 𝑁→∞.
Finally, we get 𝑟^𝑝_𝑁,𝑞=𝛿𝑓,𝑥^𝑝𝑁𝜃,𝐹^̌‌_𝑛𝑐(𝑥)𝑒^{−𝑖𝜋𝑁𝑥}−𝑒^{𝑖𝜋(𝑁+1)𝑥}+𝛿^𝑝_−𝑁𝜃,𝐹^̌‌_𝑛𝑐(𝑥)𝑒^{𝑖𝜋𝑁𝑥}−𝑒^{−𝑖𝜋(𝑁+1)𝑥}+𝑜𝑁^{−𝑞−2𝑝−1}, (52) and we need to estimate 𝛿^𝑝_±𝑁(𝜃,𝐹^̌‌_𝑛). Similarly, as above, 𝛿^𝑝_±𝑁𝜃,𝐹^̌‌_𝑛=^𝑞+2𝑝+1_𝑚=𝑞𝐴_𝑚𝑓𝛿^𝑝_±𝑁𝜃,𝐵^̌‌_𝑛(𝑚)+𝑜𝑁^{−2𝑝−𝑞−2}. (53) In view of Lemmas A.3 and A.4, 𝛿^𝑝_±𝑁𝜃,𝐹^̌‌_𝑛=𝐴_𝑞𝑓𝛿^𝑝_±𝑁𝜃,𝐵^̌‌_𝑛𝑞+𝑜𝑁^{−2𝑝−𝑞−2}=±𝐴_𝑞𝑓(−1)^𝑁+𝑝+12(𝑖𝜋)^𝑞+1𝑁^2𝑝+𝑞+1𝜓𝑞!^𝑞,𝑝+𝑂𝑁^{−2𝑝−𝑞−2}. (54) Substituting this into (52), we get the required estimate as 𝜃_𝑘→1 and lim_𝑁→∞𝑐(𝑥)=2^2𝑝+2cos^2𝑝+2𝜋𝑥2. (55)

We prove similar result for odd values of 𝑞.

Theorem 5. Let 𝑞 be odd, 𝑞≥1, and 𝑓∈𝐶^𝑞+2𝑝+2[−1,1] with 𝑓^{(𝑞+2𝑝+2)}∈𝐴𝐶[−1,1] for some 𝑝≥1. Let parameters 𝜃_𝑘 be chosen as in (32). Then, the following estimate holds for |𝑥|<1, 𝑟^𝑝_𝑁,𝑞=𝜑𝑓,𝑥^{𝑁,𝑞,𝑝}𝑓,𝑥𝑁^2𝑝+𝑞+2+𝑜𝑁^{−2𝑝−𝑞−2},𝑁→∞, (56) where 𝜑_{𝑁,𝑞,𝑝}𝑓,𝑥=𝐴_𝑞𝑓(−1)^{𝑁+𝑝+(𝑞+1)/2+1}2^2𝑝+2𝜋^𝑞+1×()(𝜋())𝑞!sin𝜋𝑥/2sin2𝑁+1𝑥/2cos^2𝑝+2()𝜓𝜋𝑥/2^𝑞+1,𝑝+𝐴_𝑞+1𝑓(−1)^{𝑁+𝑝+(𝑞+1)/2}2^2𝑝+1𝜋^𝑞+2!×(𝜋())𝑞+1sin2𝑁+1𝑥/2cos^2𝑝+1()𝜓𝜋𝑥/2^𝑞+1,𝑝. (57)

Proof. As the proof of this theorem mimics the proof of the previous one, we omit some details.
The application of transformation (14) to (17) twice with 𝜃_𝑝+1=𝜃_𝑝+2=1 implies that𝑟^𝑝_𝑁,𝑞=𝛿𝑓,𝑥^𝑝𝑁𝜃,𝐹^̌‌_𝑛𝑐(𝑥)𝑒^{−𝑖𝜋𝑁𝑥}−𝑒^{𝑖𝜋(𝑁+1)𝑥}+𝛿^𝑝_−𝑁𝜃,𝐹^̌‌_𝑛𝑐(𝑥)𝑒^{𝑖𝜋𝑁𝑥}−𝑒^{−𝑖𝜋(𝑁+1)𝑥}+𝛿¹_𝑁𝛿^𝑝_𝑛𝜃,𝐹^̌‌_𝑛𝑑(𝑥)𝑒^{−𝑖𝜋𝑁𝑥}−𝑒^{𝑖𝜋(𝑁+1)𝑥}+𝛿¹_−𝑁𝛿^𝑝_𝑛𝜃,𝐹^̌‌_𝑛𝑑(𝑥)𝑒^{𝑖𝜋𝑁𝑥}−𝑒^{−𝑖𝜋(𝑁+1)𝑥}+1𝑑(𝑥)^𝑁_𝑛=−𝑁𝛿²_𝑛𝛿^𝑝_𝑛𝜃,𝐹_𝑛−𝐹^̌‌_𝑛𝑒^{𝑖𝜋𝑛𝑥}+1𝑑(𝑥)^∞_{|𝑛|=𝑁+1}𝛿²_𝑛𝛿^𝑝_𝑛𝜃,𝐹_𝑛𝑒^{𝑖𝜋𝑛𝑥}, (58) where, by 𝛿^𝑠_𝑛(𝑦_𝑛), we denoted 𝛿^𝑠_𝑛(𝜃,𝑦_𝑛) with 𝜃_𝑘=1, 𝑘=1,…,𝑠, and 𝑐(𝑥)=4cos²𝜋𝑥2^𝑝_𝑠=11+2𝜃_𝑠cos𝜋𝑥+𝜃²_𝑠,𝑑(𝑥)=1+𝑒^{𝑖𝜋𝑥2}1+𝑒^{−𝑖𝜋𝑥2𝑝}_𝑠=11+𝜃_𝑠𝑒^𝑖𝜋𝑥1+𝜃_𝑠𝑒^{−𝑖𝜋𝑥}=16cos⁴𝜋𝑥2^𝑝_𝑠=11+2𝜃_𝑠cos𝜋𝑥+𝜃²_𝑠. (59)
We have 𝐹_𝑛=^𝑞+2𝑝+2_𝑚=𝑞𝐴_𝑚𝑓𝐵_𝑛(𝑚)+𝑜𝑛^{−2𝑝−𝑞−3},𝛿²_𝑛𝛿^𝑝_𝑛𝜃,𝐹_𝑛=^𝑞+2𝑝+2_𝑚=𝑞𝐴_𝑚𝑓𝛿²_𝑛𝛿^𝑝_𝑛𝜃,𝐵_𝑛(𝑚)+𝑜𝑛^{−2𝑝−𝑞−3}. (60) According to Lemma A.1, 𝛿²_𝑛𝛿^𝑝_𝑛𝜃,𝐹_𝑛=1𝑁^2𝑝𝑜𝑛^−𝑞−3, (61) and the last term in the right-hand side of (58) is 𝑜(𝑁^{−𝑞−2𝑝−2}) as 𝑁→∞.
Similarly, 𝛿²_𝑛𝛿^𝑝_𝑛𝜃,𝐹^̌‌_𝑛−𝐹_𝑛=^𝑞+2𝑝+2_𝑚=𝑞𝐴_𝑚𝑓𝛿²_𝑛𝛿^𝑝_𝑛𝜃,𝐵^̌‌_𝑛(𝑚)−𝐵_𝑛(𝑚)+𝑜𝑁^{−2𝑝−𝑞−3}, (62) and, according to Lemma A.2, 𝛿²_𝑛𝛿^𝑝_𝑛𝜃,𝐹^̌‌_𝑛−𝐹_𝑛=𝑜𝑁^{−2𝑝−𝑞−3},𝑁→∞. (63) Hence, the fifth term in the right-hand side of (58) is 𝑜(𝑁^{−𝑞−2𝑝−2}) as 𝑁→∞.
Then, 𝛿^𝑤_±𝑁𝛿^𝑝_𝑛𝜃,𝐹^̌‌_𝑛=^𝑞+2𝑝+2_𝑚=𝑞𝐴_𝑚𝑓𝛿^𝑤_±𝑁𝛿^𝑝_𝑛𝜃,𝐵^̌‌_𝑛(𝑚)+𝑜𝑁^{−2𝑝−𝑞−3}. (64) According to Lemma A.4, when 𝑤=1 in (64) and 𝑞 is odd, we get 𝛿¹_±𝑁𝛿^𝑝_𝑛𝜃,𝐹^̌‌_𝑛=𝐴_𝑞𝑓𝛿¹_±𝑁𝛿^𝑝_𝑛𝜃,𝐵^̌‌_𝑛𝑞+𝑜𝑁^{−2𝑝−𝑞−3}=𝑜𝑁^{−2𝑝−𝑞−3}, (65) and hence the third and fourth terms in the right hand side of (58) are 𝑜(𝑁^{−𝑞−2𝑝−2}) as 𝑁→∞. Now, from (58), we derive 𝑟^𝑝_𝑁,𝑞=𝛿𝑓,𝑥^𝑝𝑁𝜃,𝐹^̌‌_𝑛𝑐(𝑥)𝑒^{−𝑖𝜋𝑁𝑥}−𝑒^{𝑖𝜋(𝑁+1)𝑥}+𝛿^𝑝_−𝑁𝜃,𝐹^̌‌_𝑛𝑐(𝑥)𝑒^{𝑖𝜋𝑁𝑥}−𝑒^{−𝑖𝜋(𝑁+1)𝑥}+𝑜𝑁^{−𝑞−2𝑝−2}. (66) Taking 𝑤=0 in (64), we write 𝛿^𝑝_±𝑁𝜃,𝐹^̌‌_𝑛=^𝑞+2𝑝+2_𝑚=𝑞𝐴_𝑚𝑓𝛿^𝑝_±𝑁𝜃,𝐵^̌‌_𝑛(𝑚)+𝑜𝑁^{−2𝑝−𝑞−3}=𝐴_𝑞𝑓𝛿^𝑝_±𝑁𝜃,𝐵^̌‌_𝑛𝑞+𝐴_𝑞+1𝑓𝛿^𝑝_±𝑁𝜃,𝐵^̌‌_𝑛𝑁𝑞+1+𝑜^{−2𝑝−𝑞−3}. (67) Now, the application of Lemmas A.2 and A.3 completes the proof.

Note that Theorems 4 and 5 are valid also for 𝑝=0. In that case, the rtp-interpolation coincides with the KL-interpolation as 𝐼⁰_𝑁,𝑞(𝑓,𝑥)≡𝐼_𝑁,𝑞(𝑓,𝑥) and consequently 𝑟⁰_𝑁,𝑞(𝑓,𝑥)≡𝑟_𝑁,𝑞(𝑓,𝑥). Hence, for 𝑝=0, Theorems 4 and 5 coincide with Theorems 1 and 2, respectively.

5. Results and Discussion

First, let us compare Theorems 4 and 5. Theorem 4 states that on the interval |𝑥|<1, the rate of convergence of 𝐼^𝑝_𝑁,𝑞(𝑓,𝑥) is 𝑂(𝑁^{−𝑞−2𝑝−1}) for even values of 𝑞. According to Theorem 5, the rate of convergence of 𝐼^𝑝_𝑁,𝑞(𝑓,𝑥) for odd values of 𝑞 is 𝑂(𝑁^{−𝑞−2𝑝−2}), and we have an improvement in the convergence by the factor 𝑂(𝑁). From the other side, Theorem 5 puts an additional smoothness requirement on the interpolated function. Moreover, while the estimate in Theorem 4 depends only on 𝐴_𝑞(𝑓), the estimate of Theorem 5 depends on both 𝐴_𝑞(𝑓) and 𝐴_𝑞+1(𝑓). All these mimic the behavior of the KL-interpolation where we have the same differences in the asymptotic estimates of Theorems 1 and 2.

Now, let us compare convergence of the KL- and the rtp-interpolations. In this section, we suppose that parameters 𝜏_𝑘 in (32) are the roots of the associated Laguerre polynomial 𝐿^𝑞_𝑝(𝑥). Theorems concerning the rtp-interpolations put additional smoothness requirements on the interpolated function, and, in comparisons, it must be taken into account.

If an interpolated function is rather smooth (e.g., when function is infinitely differentiable) such that for given 𝑞 and 𝑝 Theorems 4 and 5 are valid, then the rtp-interpolation is more precise (however, asymptotically) than the KL-interpolation. Comparison of Theorems 1 and 2 with Theorems 4 and 5 shows that the rtp-interpolation is more precise than the KL-interpolation by factor 𝑂(𝑁^2𝑝).

Now, we consider functions with finite smoothness and suppose that all needed jumps are exactly available. Estimates proved above show (see factors 𝐴_𝑞(𝑓) and 𝐴_𝑞+1(𝑓)) that the utilization of all available jumps by the KL-interpolation is not reasonable when the jumps of the interpolated function are rapidly increasing. In such cases, more accuracy can be achieved with less jumps in combination with rational corrections.

Further in this section, we suppose that parameters 𝜏_𝑘 are the roots of the associated Laguerre polynomials 𝐿^𝑞_𝑝(𝑥).

Let 𝑓∈𝐶^𝑀+1[−1,1], 𝑓^(𝑀+1)∈𝐴𝐶[−1,1], 𝑀≥1, and let 𝑞 be an even number. According to Theorems 1 and 4, if the values of 𝑝 and 𝑞 satisfy the condition 𝑞+2𝑝=𝑀, then both theorems are valid, and comparison of corresponding approximations is legal. Then, asymptotic estimates of these theorems will show which values of parameters 𝑝 and 𝑞 provide with better accuracy. We show this process for a specific example. Let 𝑓(𝑥)=sin(𝑎𝑥−1), (68) where 𝑎 is some parameter. We use the values 𝑎=1/10,1,10,30,50, and in Table 1 to calculate the values of 𝑁^{−𝑞−2𝑝−1}max_{𝑥∈[−0.5,0.5]}|𝜑_{𝑁,𝑞,𝑝}(𝑓,𝑥)| for 𝑁=1024 and for different values of 𝑝 and 𝑞 with condition 𝑞+2𝑝=8. Recall that 𝑝=0 corresponds to the KL-interpolation.

Table 1 

Values of 𝑁^{−𝑞−2𝑝−1}max_{𝑥∈[−0.5,0.5]}|𝜑_{𝑁,𝑞,𝑝}(𝑓,𝑥)| (see (38)) for 𝑁=1024, 𝑞+2𝑝=8 and for function (68). Parameters 𝜏_𝑘 are the roots of the associated Laguerre polynomials 𝐿^𝑞_𝑝(𝑥).

Table 1 shows that for 𝑎=1/10 and 𝑎=1 utilization of all available jumps is reasonable and the KL-interpolation 𝐼_𝑁,8(𝑓,𝑥) has the best accuracy. For 𝑎=10,30, and 50, the rtp-interpolation 𝐼³_𝑁,2(𝑓,𝑥) is the best, and as larger is the value of parameter 𝑎 as more precise is the rtp-interpolation compared to the KL-interpolation.

It is important to note that results in Table 1 are obtained based on asymptotic estimates and we must compare these results to actual errors max_{𝑥∈[−0.5,0.5]}|𝑟^𝑝_𝑁,𝑞(𝑓,𝑥)| to see how the asymptotic errors coincide with actual errors. In Table 2, the values of the actual errors max_{𝑥∈[−0.5,0.5]}|𝑟^𝑝_𝑁,𝑞(𝑓,𝑥)| for 𝑁=1024, and for different values of 𝑝 and 𝑞 with condition 𝑞+2𝑝=8 are calculated. Comparison with Table 1 shows that for 𝑁=1024 theoretical estimates of Theorems 1 and 4 coincide with actual errors rather precisely.

Table 2 

Values of max_{𝑥∈[−0.5,0.5]}|𝑟^𝑝_𝑁,𝑞(𝑓,𝑥)| (see (26)) for 𝑁=1024, 𝑞+2𝑝=8, and for function (68). Parameters 𝜏_𝑘 are the roots of the associated Laguerre polynomials 𝐿^𝑞_𝑝(𝑥).

We have the same situation for odd values of 𝑞. Let 𝑓∈𝐶^𝑀+2[−1,1], 𝑓^(𝑀+2)∈𝐴𝐶[−1,1], 𝑀≥1, and let 𝑞 be an odd number. We use the same values of 𝑎 and in Table 3 to calculate the values of (1/𝑁^𝑞+2𝑝+2)max_{𝑥∈[−0.5,0.5]}|𝜑_{𝑁,𝑞,𝑝}(𝑓,𝑥)| and in Table 4 to calculate the values of max_{𝑥∈[−0.5,0.5]}|𝑟^𝑝_𝑁,𝑞(𝑓)| for 𝑁=1024 and for different values of 𝑝 and 𝑞 with condition 𝑞+2𝑝=7 when both Theorems 2 and 5 are valid. Again, for 𝑎=1/10 and 𝑎=1, the best accuracy has the KL-interpolation. For 𝑎=10,𝑎=30, and 𝑎=50, the best is rtp-interpolation 𝐼³_𝑁,1(𝑓,𝑥).

Table 3 

Values of 𝑁^{−2𝑝−𝑞−2}max_{𝑥∈[−0.5,0.5]}|𝜑_{𝑁,𝑞,𝑝}(𝑓,𝑥)| (see (57)) for 𝑁=1024, 𝑞+2𝑝=7 and for function (68). Parameters 𝜏_𝑘 are the roots of the associated Laguerre polynomials 𝐿^𝑞_𝑝(𝑥).

Table 4 

Values of max_{𝑥∈[−0.5,0.5]}|𝑟^𝑝_𝑁,𝑞(𝑓,𝑥)| (see (26)) for 𝑁=1024, 𝑞+2𝑝=7 and for function (68). Parameters 𝜏_𝑘 are the roots of the associated Laguerre polynomials 𝐿^𝑞_𝑝(𝑥).

Overall conclusion based on these specific examples and on comparison of the asymptotic estimates is the following: not always the utilization of all available jumps, by the KL-interpolation, leads to the best interpolation (if we mean pointwise convergence in the regions away from the singularities where the rtp-interpolation has faster convergence rate.) This is due to the factors 𝐴_𝑞(𝑓) and 𝐴_𝑞+1(𝑓) in the estimates. When the values of jumps are rapidly increasing, then better accuracy can be achieved by the utilization of smaller number of jumps (consequently, with smaller 𝐴_𝑞(𝑓) and 𝐴_𝑞+1(𝑓)) and appropriately chosen corrections based on the smoothness of the interpolated function. Which choice of 𝑞 and 𝑝 is the best that can be concluded from the comparison of the corresponding estimates as we did it above.

It must be also mentioned that when the jumps are rapidly increasing then getting their approximations is problematic, so in such cases, the utilization of the rational corrections is unavoidable for better accuracy.

Appendix

In this section, we prove some lemmas concerning generalized finite differences 𝛿^𝑠_𝑛(𝜃,𝑦_𝑛). By 𝛿^𝑠_𝑛(𝑦_𝑛), we denote the sequence that corresponds to 𝜃_𝑘=1, 𝑘=1,…,𝑠. By Δ^𝑘_𝑛(𝑐_𝑛), we denote the differences defined by the relations Δ⁰_𝑛𝑐_𝑛=𝑐_𝑛,Δ^𝑘_𝑛𝑐_𝑛=Δ_𝑛^𝑘−1𝑐_𝑛+Δ^𝑘−1_𝑛−1𝑐_𝑛. (A.1) Now, it is easy to verify that 𝛿^𝑘_𝑛𝑐_𝑛=Δ^2𝑘_𝑛+𝑘𝑐_𝑛. (A.2)

Lemma A.1. The following estimate holds for 𝑝,𝑤,𝑚≥0 as 𝑁→∞ and |𝑛|≥𝑁+1, where 𝜃_𝑘 are chosen as in (32) 𝛿^𝑤_𝑛𝛿^𝑝_𝑛𝜃,𝐵_𝑛(𝑚)=(−1)^{𝑛+𝑝+𝑤+1}2(𝑖𝜋𝑛)^𝑚+1𝑛^2𝑤+2𝑝×𝑚!_𝑝𝑠=0(−1)^𝑠𝛾_𝑠(𝜏)𝑁^𝑠𝑛^𝑝−𝑠_𝑘=0𝛾_𝑘(𝜏)𝑁^𝑘𝑛^−𝑘×!+12𝑤+2𝑝−𝑘−𝑠+𝑚𝑁^2𝑝𝑂𝑛^{−2𝑤−𝑚−2}. (A.3)

Proof. In view of definitions of 𝛿^𝑘_𝑛(𝜃,𝑦_𝑛), 𝛿^𝑘_𝑛(𝑦_𝑛) and their connection with Δ^𝑘_𝑛(𝑦_𝑛), we write 𝛿^𝑝_𝑛𝜃,𝐵_𝑛(𝑚)=^𝑝_𝑠=0(−1)^𝑠𝛾_𝑠(𝜏)𝑁^𝑠𝑝_𝑘=0(−1)^𝑘𝛾_𝑘(𝜏)𝑁^𝑘Δ^{2𝑝−𝑘−𝑠}_{𝑛+𝑝−𝑠}𝐵_𝑛(𝑚). (A.4) Then, 𝛿^𝑤_𝑛𝛿^𝑝_𝑛𝜃,𝐵_𝑛(𝑚)=^𝑝_𝑠=0(−1)^𝑠𝛾_𝑠(𝜏)𝑁^𝑠𝑝_𝑘=0(−1)^𝑘𝛾_𝑘(𝜏)𝑁^𝑘Δ^2𝑤_𝑛+𝑤Δ^{2𝑝−𝑘−𝑠}_{𝑛+𝑝−𝑠}𝐵_𝑛(𝑚). (A.5) Taking into account that Δ^2𝑤_𝑛+𝑤Δ^{2𝑝−𝑘−𝑠}_{𝑛+𝑝−𝑠}𝐵_𝑛(𝑚)=Δ^{2𝑤+2𝑝−𝑘−𝑠}_{𝑛+𝑝−𝑠+𝑤}𝐵_𝑛(𝑚), (A.6) we find 𝛿^𝑤_𝑛𝛿^𝑝_𝑛𝜃,𝐵_𝑛(𝑚)=^𝑝_𝑠=0(−1)^𝑠𝛾_𝑠(𝜏)𝑁^𝑠𝑝_𝑘=0(−1)^𝑘𝛾_𝑘(𝜏)𝑁^𝑘Δ^{2𝑤+2𝑝−𝑘−𝑠}_{𝑛+𝑝−𝑠+𝑤}𝐵_𝑛(𝑚). (A.7) Moreover, since Δ^𝑤_𝑛𝑦_𝑛=^2𝑤_ℓ=0ℓ𝑦2𝑤_𝑛−ℓ, (A.8) we obtain 𝛿^𝑤_𝑛𝛿^𝑝_𝑛𝜃,𝐵_𝑛(𝑚)=^𝑝_𝑠=0(−1)^𝑠𝛾_𝑠(𝜏)𝑁^𝑠𝑝_𝑘=0(−1)^𝑘𝛾_𝑘(𝜏)𝑁^𝑘×^{2𝑤+2𝑝−𝑘−𝑠}_ℓ=0ℓ𝐵2𝑤+2𝑝−𝑘−𝑠_{𝑛+𝑝−𝑠+𝑤−ℓ}(𝑚). (A.9)
Taking into account the explicit form of the Fourier coefficients of the Bernoulli polynomials 𝐵_{𝑛+𝑝−𝑠+𝑤−ℓ}(𝑚)=(−1)^{𝑛+𝑝−𝑠+𝑤−ℓ+1}2𝑖𝜋𝑛+𝑝−𝑠+𝑤−ℓ^𝑚+1=(−1)^{𝑛+𝑝+𝑠+𝑤+ℓ+1}2(𝑖𝜋𝑛)^𝑚+1×^∞_𝑗=0𝑚𝑗+𝑚ℓ+𝑠−𝑤−𝑝^𝑗𝑛^𝑗=(−1)^{𝑛+𝑝+𝑠+𝑤+ℓ+1}2(𝑖𝜋𝑛)^∞𝑚+1_𝑗=01𝑛^𝑗𝑚×𝑗+𝑚_𝑗𝑡=0𝑗𝑡ℓ^𝑡𝑠−𝑤−𝑝^𝑗−𝑡, (A.10) we derive 𝛿^𝑤_𝑛𝛿^𝑝_𝑛𝜃,𝐵_𝑛(𝑚)=(−1)^{𝑛+p+𝑤+1}2(𝑖𝜋𝑛)^𝑝𝑚+1_𝑠=0𝛾_𝑠(𝜏)𝑁^𝑠𝑝_𝑘=0(−1)^𝑘𝛾_𝑘(𝜏)𝑁^𝑘×^∞_𝑗=01𝑛^𝑗𝑚×𝑗+𝑚_𝑗𝑡=0𝑗𝑡𝑠−𝑤−𝑝^𝑗−𝑡𝛼_{2𝑤+2𝑝−𝑘−𝑠,𝑡}, (A.11) where 𝛼_𝑟,𝑡=^𝑟_ℓ=0(−1)^ℓ𝑟ℓℓ^𝑡. (A.12)
Note that 𝛼_𝑟,𝑡=0,𝑡≤𝑟−1. (A.13)
Thus, 𝛿^𝑤_𝑛𝛿^𝑝_𝑛𝜃,𝐵_𝑛(𝑚)=(−1)^{𝑛+𝑝+𝑤+1}2(𝑖𝜋𝑛)^𝑚+1𝑛^{𝑝2𝑤+2𝑝}_𝑠=0𝛾_𝑠(𝜏)𝑁^𝑠𝑝_𝑘=0(−1)^𝑘𝛾_𝑘(𝜏)𝑁^𝑘×^∞_𝑗=01𝑛^{𝑗−𝑘−𝑠}𝑚×2𝑤+2𝑝−𝑘−𝑠+𝑗+𝑚_𝑗𝑡=0𝑗+2𝑤+2𝑝−𝑘−𝑠𝑡+2𝑤+2𝑝−𝑘−𝑠𝑠−𝑤−𝑝^𝑗−𝑡×𝛼_{2𝑤+2𝑝−𝑘−𝑠,𝑡+2𝑤+2𝑝−𝑘−𝑠}=(−1)^{𝑛+𝑝+𝑤+1}2(𝑖𝜋𝑛)^𝑚+1𝑛^2𝑤+2𝑝𝑚!^𝑝_𝑠=0𝛾_𝑠(𝜏)𝑁^𝑠𝑛^𝑝−𝑠_𝑘=0(−1)^𝑘𝛾_𝑘(𝜏)𝑁^𝑘𝑛^−𝑘×𝑚𝛼2𝑤+2𝑝−𝑘−𝑠+𝑚_{2𝑤+2𝑝−𝑘−𝑠,2𝑤+2𝑝−𝑘−𝑠}+1𝑁^2𝑝𝑂𝑛^{−2𝑤−𝑚−2}, (A.14) which completes the proof as 𝛼_{2𝑤+2𝑝−𝑘−𝑠,2𝑤+2𝑝−𝑘−𝑠}=(−1)^{2𝑤+2𝑝−𝑘−𝑠}2𝑤+2𝑝−𝑘−𝑠!. (A.15)

The proof of next lemmas can be obtained in a similar manner.

Lemma A.2. The following estimate holds for 𝑝,𝑤,𝑚≥0 as 𝑁→∞ and |𝑛|≤𝑁, where 𝜃_𝑘 are chosen as in (32) 𝛿^𝑤_𝑛𝛿^𝑝_𝑛𝜃,𝐵^̌‌_𝑛(𝑚)−𝐵_𝑛(𝑚)=(−1)^{𝑛+𝑝+𝑤+1}2(𝑖𝜋𝑁)^𝑚+1𝑁^2𝑤+2𝑝𝑚!^𝑝_𝑠=0(−1)^𝑠𝛾_𝑠(𝜏)^𝑝_𝑘=0𝛾_𝑘(𝜏)×2𝑤+2𝑝−𝑘−𝑠+𝑚!×_𝑟≠0(−1)^𝑟(2𝑟+𝑛/𝑁)^{2𝑤+2𝑝−𝑘−𝑠+𝑚+1}+𝑂𝑁^{−2𝑤−2𝑝−𝑚−2}. (A.16)

Proof. Similar to (A.9), we get 𝛿^𝑤_𝑛𝛿^𝑝_𝑛𝜃,𝐵^̌‌_𝑛(𝑚)−𝐵_𝑛(𝑚)=^𝑝_𝑠=0(−1)^𝑠𝛾_𝑠(𝜏)𝑁^𝑠𝑝_𝑘=0(−1)^𝑘𝛾_𝑘(𝜏)𝑁^𝑘×^{2𝑤+2𝑝−𝑘−𝑠}_ℓ=0ℓ×2𝑤+2𝑝−𝑘−𝑠_̌‌𝐵_{𝑛+𝑝−𝑠+𝑤−ℓ}(𝑚)−𝐵_{𝑛+𝑝−𝑠+𝑤−ℓ}(𝑚). (A.17) Taking into account that 𝐵^̌‌_𝑛(𝑚)−𝐵_𝑛(𝑚)=_𝑟≠0𝐵_{𝑛+𝑟(2𝑁+1)}(𝑚)=(−1)^𝑛+12(𝑖𝜋)^𝑚+1_𝑟≠0(−1)^𝑟(𝑛+𝑟(2𝑁+1))^𝑚+1, (A.18) we derive 𝛿^𝑤_𝑛𝛿^𝑝_𝑛𝜃,𝐵^̌‌_𝑛(𝑚)−𝐵_𝑛(𝑚)=(−1)^{𝑛+𝑝+𝑤+1}2(𝑖𝜋𝑁)^𝑝𝑚+1_𝑠=0𝛾_𝑠(𝜏)𝑁^𝑠𝑝_𝑘=0(−1)^𝑘𝛾_𝑘(𝜏)𝑁^k×^∞_𝑗=01𝑁^𝑗𝑚𝑗+𝑚^𝑗_𝑡=0𝑗𝑡𝛼_{2𝑤+2𝑝−𝑘−𝑠,𝑡}×_𝑟≠0(−1)^𝑟𝑠−𝑤−𝑝−𝑟^𝑗−𝑡(2𝑟+𝑛/𝑁)^𝑗+𝑚+1. (A.19) In view of identity (A.13), we have 𝛿^𝑤_𝑛𝛿^𝑝_𝑛𝜃,𝐵^̌‌_𝑛(𝑚)−𝐵_𝑛(𝑚)=(−1)^{𝑛+𝑝+𝑤+1}2(𝑖𝜋𝑁)^𝑚+1𝑁^{𝑝2𝑤+2𝑝}_𝑠=0𝛾_𝑠(𝜏)^𝑝_𝑘=0(−1)^𝑘𝛾_𝑘(𝜏)×^∞_𝑗=01𝑁^𝑗𝑚×𝑗+2𝑤+2𝑝−𝑘−𝑠+𝑚_𝑗𝑡=0𝑗+2𝑤+2𝑝−𝑘−𝑠𝑡+2𝑤+2𝑝−𝑘−𝑠×𝛼_{2𝑤+2𝑝−𝑘−𝑠,𝑡+2𝑤+2𝑝−𝑘−𝑠𝑟≠0}(−1)^𝑟𝑠−𝑤−𝑝−𝑟^𝑗−𝑡(2𝑟+𝑛/𝑁)^{𝑗+2𝑤+2𝑝−𝑘−𝑠+𝑚+1}=(−1)^{𝑛+𝑝+𝑤+1}2(𝑖𝜋𝑁)^𝑚+1𝑁^2𝑤+2𝑝×^𝑝_𝑠=0𝛾_𝑠(𝜏)^𝑝_𝑘=0(−1)^𝑘𝛾_𝑘(𝜏)𝑚2𝑤+2𝑝−𝑘−𝑠+𝑚×𝛼_{2𝑤+2𝑝−𝑘−𝑠,2𝑤+2𝑝−𝑘−𝑠}×_𝑟≠0(−1)^𝑟(2𝑟+𝑛/𝑁)^{2𝑤+2𝑝−𝑘−𝑠+𝑚+1}+𝑂𝑁^{−2𝑤−2𝑝−𝑚−2}, (A.20) which completes the proof as 𝛼_{2𝑤+2𝑝−𝑘−𝑠,2𝑤+2𝑝−𝑘−𝑠}=(−1)^{2𝑤+2𝑝+𝑘+𝑠}2𝑤+2𝑝−𝑘−𝑠!. (A.21)

Lemma A.3. Let 𝑚 be even. Then, the following estimate holds for 𝑝,𝑤,𝑚≥0 as 𝑁→∞, where 𝜃_𝑘 are chosen as in (32) 𝛿^𝑤_±𝑁𝛿^𝑝_𝑛𝜃,𝐵^̌‌_𝑛(𝑚)=(−1)^{𝑁+𝑝+𝑤+1}2(𝑖𝜋𝑁)^𝑚+1𝑁^2𝑤+2𝑝𝑚!^𝑝_𝑠=0(−1)^𝑠𝛾_𝑠(𝜏)^𝑝_𝑘=0𝛾_𝑘(𝜏)×2𝑤+2𝑝−𝑘−𝑠+𝑚!^∞_𝑟=−∞(−1)^𝑟(2𝑟±1)^{2𝑤+2𝑝−𝑘−𝑠+𝑚+1}+𝑂𝑁^{−2𝑤−2𝑝−𝑚−2}. (A.22)

Proof. Similar to (A.9), we get 𝛿^𝑤_±𝑁𝛿^𝑝_𝑛𝜃,𝐵^̌‌_𝑛(𝑚)=^𝑝_𝑠=0(−1)^𝑠𝛾_𝑠(𝜏)𝑁^𝑠𝑝_𝑘=0(−1)^𝑘𝛾_𝑘(𝜏)𝑁^𝑘×^{2𝑤+2𝑝−𝑘−𝑠}_ℓ=0ℓ2𝑤+2𝑝−𝑘−𝑠_̌‌𝐵_{±𝑁+𝑝−𝑠+𝑤−ℓ}(𝑚). (A.23) Taking into account that 𝐵^̌‌_𝑛(𝑚)=^∞_𝑠=−∞𝐵_{𝑛+𝑠(2𝑁+1)}(𝑚)=(−1)^𝑛+12(𝑖𝜋)^∞𝑚+1_𝑠=−∞(−1)^𝑠(𝑛+𝑠(2𝑁+1))^𝑚+1, (A.24) we have 𝐵^̌‌_{±𝑁+𝑝−𝑠+𝑤−ℓ}(𝑚)=(−1)^{𝑁+𝑝+𝑤+𝑠+ℓ+1}2(𝑖𝜋𝑁)^∞𝑚+1_𝑗=01𝑁^𝑗𝑚×𝑗+𝑚_{∞𝑟=−∞}(−1)^𝑟ℓ+𝑠−𝑝−𝑤−𝑟^𝑗(2𝑟±1)^𝑚+𝑗+1. (A.25) Hence, 𝛿^𝑤_±𝑁𝛿^𝑝_𝑛𝜃,𝐵^̌‌_𝑛(𝑚)=(−1)^{𝑁+𝑝+𝑤+1}2(𝑖𝜋𝑁)^𝑚+1𝑁^{𝑝2𝑤+2𝑝}_𝑠=0𝛾_𝑠(𝜏)^𝑝_𝑘=0(−1)^𝑘𝛾_𝑘(𝜏)×^∞_𝑗=01𝑁^𝑗𝑚×𝑗+2𝑤+2𝑝−𝑘−𝑠+𝑚_𝑗𝑡=0𝛼𝑗+2𝑤+2𝑝−𝑘−𝑠𝑡+2𝑤+2𝑝−𝑘−𝑠^{2𝑤+2𝑝−𝑘−𝑠,𝑡+2𝑤+2𝑝−𝑘−𝑠}×^∞_𝑟=−∞(−1)^𝑟𝑠−𝑝−𝑤−𝑟^𝑗−𝑡(2𝑟±1)^{𝑗+2𝑤+2𝑝−𝑘−𝑠+𝑚+1}. (A.26) From here, we get 𝛿^𝑤_±𝑁𝛿^𝑝_𝑛𝜃,𝐵^̌‌_𝑛(𝑚)=(−1)^{𝑁+𝑝+𝑤+1}2(𝑖𝜋𝑁)^𝑚+1𝑁^{𝑝2𝑤+2𝑝}_𝑠=0𝛾_𝑠(𝜏)^𝑝_𝑘=0(−1)^𝑘𝛾_𝑘(𝜏)×𝑚𝛼2𝑤+2𝑝−𝑘−𝑠+𝑚_{2𝑤+2𝑝−𝑘−𝑠,2𝑤+2𝑝−𝑘−𝑠}×^∞_𝑟=−∞(−1)^𝑟(2𝑟±1)^{2𝑤+2𝑝−𝑘−𝑠+𝑚+1}+𝑂𝑁^{−2𝑤−2𝑝−𝑚−2}, (A.27) which completes the proof.

Lemma A.4. Let 𝑚 be odd. Then, the following estimate holds for 𝑝,𝑤≥0 and 𝑚≥1 as 𝑁→∞, where 𝜃_𝑘 are chosen as in (32) 𝛿^𝑤_±𝑁𝛿^𝑝_𝑛𝜃,𝐵^̌‌_𝑛(𝑚)=(−1)^{𝑁+𝑝+𝑤}2(𝑖𝜋𝑁)^𝑚+1𝑁^{2𝑤+2𝑝+1}𝑚!^𝑝_𝑠=0(−1)^𝑠𝛾_𝑠(𝜏)^𝑝_𝑘=0𝛾_𝑘(𝜏)×!×2𝑤+2𝑝−𝑘−𝑠+𝑚+1_{∞𝑟=−∞}(−1)^𝑟𝑟(2𝑟±1)^{2𝑤+2𝑝−𝑘−𝑠+𝑚+2}+𝑂𝑁^{−2𝑤−2𝑝−𝑚−3}. (A.28)

Proof. We again use (A.26) and explore terms 𝑗=0 and 𝑗=1𝛿^𝑤_±𝑁𝛿^𝑝_𝑛𝜃,𝐵^̌‌_𝑛(𝑚)=(−1)^{𝑁+𝑝+𝑤+1}2(𝑖𝜋𝑁)^𝑚+1𝑁^2𝑤+2𝑝𝑚!^𝑝_𝑠=0(−1)^𝑠𝛾_𝑠(𝜏)^𝑝_𝑘=0𝛾_𝑘(𝜏)×2𝑤+2𝑝−𝑘−𝑠+𝑚!^∞_𝑟=−∞(−1)^𝑟(2𝑟±1)^{2𝑤+2𝑝−𝑘−𝑠+𝑚+1}+(−1)^{𝑁+𝑝+𝑤+1}4(𝑖𝜋𝑁)^𝑚+1𝑁^{2𝑤+2𝑝+1}𝑚!^𝑝_𝑠=0(−1)^𝑠𝛾_𝑠(𝜏)^𝑝_𝑘=0𝛾_𝑘(𝜏)×!()×2𝑤+2𝑝−𝑘−𝑠+𝑚+1𝑠−𝑘_{∞𝑟=−∞}(−1)^𝑟(2𝑟±1)^{2𝑤+2𝑝−𝑘−𝑠+𝑚+2}+(−1)^{𝑁+𝑝+𝑤}2(𝑖𝜋𝑁)^𝑚+1𝑁^{2𝑤+2𝑝+1}𝑚!^𝑝_𝑠=0(−1)^𝑠𝛾_𝑠(𝜏)^𝑝_𝑘=0𝛾_𝑘(𝜏)×!×2𝑤+2𝑝−𝑘−𝑠+𝑚+1_{∞𝑟=−∞}(−1)^𝑟𝑟(2𝑟±1)^{2𝑤+2𝑝−𝑘−𝑠+𝑚+2}+𝑂𝑁^{−2𝑤−2𝑝−𝑚−3},𝑁→∞. (A.29) It is easy to verify that the first two terms in the right-hand side of (A.29) vanish for odd values of 𝑚. This observation completes the proof.

Notations

𝑓𝑛:	The Fourier coefficient (see (5))
𝑓̌‌𝑛:	The discrete Fourier coefficient (see (2))
𝐼𝑁(𝑓,𝑥):	Classic trigonometric interpolation (see (1))
𝑟𝑁(𝑓,𝑥):	Error of classic trigonometric interpolation (see (4))
𝐼𝑝𝑁(𝑓,𝑥):	Rational-trigonometric interpolation (see (16))
𝑟𝑝𝑁(𝑓,𝑥):	Error of rational-trigonometric interpolation (see (17))
𝐼𝑁,𝑞(𝑓,𝑥):	The Krylov-Lanczos interpolation (see (23))
𝑟𝑁,𝑞(𝑓,𝑥):	Error of the Krylov-Lanczos interpolation (see (25))
𝐼𝑝𝑁,𝑞(𝑓,𝑥):	Rational-trigonometric-polynomial interpolation (see (24))
𝑟𝑝𝑁,𝑞(𝑓,𝑥):	Error of rational-trigonometric-polynomial interpolation (see (26))
𝐴𝑘(𝑓):	Jumps of 𝑓 (see (18))
𝛿𝑠𝑛(𝜃,𝑦𝑛):	Generalized finite difference (see (13))
𝛿𝑠𝑛(𝑦𝑛):	Generalized finite difference with 𝜃𝑘=1, 𝑘=1,…,𝑠   (see Section 4 and the appendix)
Δ𝑠𝑛(𝑦𝑛):	Finite difference (see (A.1))
𝐵𝑘(𝑥):	The Bernoulli polynomial (see (20))
𝐵𝑛(𝑘):	The Fourier coefficient of the Bernoulli polynomial (see (21))
𝜏𝑘:	Some parameters joined to 𝜃𝑘 (see (32))
𝛾𝑠(𝜏):	Coefficients of polynomial with the roots −1/𝜏𝑘 (see (35)).

Acknowledgment

The author acknowledges the reviewers and the editors for their valuable suggestions and constructive comments that helped to improve the paper.