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

Authors: Arnak Poghosyan
Journal: Advances in Numerical Analysis
 (2013)

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## 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.


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## Body

## 1. Introduction

In this paper, we continue investigations started in [1] where we considered the convergence acceleration of the classical trigonometric interpolation
(1)IN(f,x)=∑n=-NN‍fˇneiπnx,(2)fˇn=12N+1∑k=-NN‍f(xk)e-iπnxk,xk=2k2N+1,
via polynomial corrections representing discontinuities of the function f and some of its first q derivatives (jumps). The resultant interpolation IN,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 (byeiπx) 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,…,θp) rational-trigonometric-polynomial (rtp-) interpolation IN,qp(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 θk=0, k=1,…,p. Besides, rational corrections can be applied immediately to the classical interpolation without polynomial corrections (see interpolation INp(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].ByrN(f,x), we denote the error of the classical trigonometric interpolation
(3)rN(f,x)=f(x)-IN(f,x),
and write
(4)rN(f,x)=∑n=-NN‍(fn-fˇn)eiπnx+∑n=N+1∞‍fneiπnx+∑n=-∞-N-1‍fneiπnx,
where fn is the nth Fourier coefficient of f(5)fn=12∫-11‍f(x)e-iπnxdx.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 eiπx) functions. Consider a vector of complex numbers θ={θ1,…,θp}. The first formula of summation by parts is easy to verify straightforwardly as
(6)∑n=N+1∞‍fneiπnx=-θ1fNeiπ(N+1)x1+θ1eiπx+11+θ1eiπx∑n=N+1∞‍(fn+θ1fn-1)eiπnx.
If here |θ1|<1, then the formula is valid for all |x|≤1. This is the main reason of including the parameters θk 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
(7)∑n=N+1∞‍fneiπnx=θ1fN+1eiπNx1+θ1e-iπx+11+θ1e-iπx∑n=N+1∞‍(fn+θ1fn+1)eiπnx.
Application of (7) to the second term in the right-hand side of (6) leads to the needed expansion
(8)∑n=N+1∞‍fneiπnx=1(1+θ1eiπx)(1+θ1e-iπx)×∑n=N+1∞(fn+θ1fn-1+θ1(fn+1+θ1fn))eiπnx-θ1fNeiπ(N+1)x(1+θ1eiπx)(1+θ1e-iπx)+θ1fN+1eiπNx(1+θ1eiπx)(1+θ1e-iπx).
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):
(9)∑n=-∞-N-1‍fneiπnx=1(1+θ1eiπx)(1+θ1e-iπx)×∑n=-∞-N-1(fn+θ1fn-1+θ1(fn+1+θ1fn))eiπnx-θ1f-Ne-iπ(N+1)x(1+θ1eiπx)(1+θ1e-iπx)+θ1f-N-1e-iπNx(1+θ1eiπx)(1+θ1e-iπx).For the first term in the right-hand side of (4), the formula of summation by parts is the following:
(10)∑n=-NN‍cneiπnx=1(1+θ1eiπx)(1+θ1e-iπx)×∑n=-NN‍(cn+θ1cn-1+θ1(cn+1+θ1cn))eiπnx+θ1cNeiπ(N+1)x(1+θ1eiπx)(1+θ1e-iπx)-θ1c-N-1e-iπNx(1+θ1eiπx)(1+θ1e-iπx)-θ1cN+1eiπNx(1+θ1eiπx)(1+θ1e-iπx)+θ1c-Ne-iπ(N+1)x(1+θ1eiπx)(1+θ1e-iπx),
where cn=fn-fˇn.Substituting (8), (9), and (10) into (4), after some simplifications, provides with the following expansion of the error
(11)rN(f,x)=θ1fˇNe-iπNx-eiπ(N+1)x(1+θ1eiπx)(1+θ1e-iπx)+θ1fˇ-NeiπNx-e-iπ(N+1)x(1+θ1eiπx)(1+θ1e-iπx)+1(1+θ1eiπx)(1+θ1e-iπx)×∑|n|=N+1∞‍(fn+θ1fn-1+θ1(fn+1+θ1fn))eiπnx+1(1+θ1eiπx)(1+θ1e-iπx)×∑n=-NN‍(cn+θ1cn-1+θ1(cn+1+θ1cn))eiπnx,
where cn=fn-fˇn. Here, we also took into account the periodicity of the coefficients fˇn(12)fˇN+1=fˇ-N,fˇ-N-1=fˇN.
For writing the expansion (11) in a short form and also for further reiterations of this transformation, we introduce the following generalized finite differences δnk(θ,yn) determined recurrently:
(13)δn0(θ,yn)=yn,δnk(θ,yn)=δnk-1(θ,yn)+θkδn-1k-1(θ,yn)+θk(δn+1k-1(θ,yn)+θkδnk-1(θ,yn)),
for some sequence yn. Now, (11) can be rewritten in the form
(14)rN(f,x)=θ1δN0(θ,fˇn)e-iπNx-eiπ(N+1)x(1+θ1eiπx)(1+θ1e-iπx)+θ1δ-N0(θ,fˇn)eiπNx-e-iπ(N+1)x(1+θ1eiπx)(1+θ1e-iπx)+1(1+θ1eiπx)(1+θ1e-iπx)×∑|n|=N+1∞‍δn1(θ,fn)eiπnx+1(1+θ1eiπx)(1+θ1e-iπx)×∑n=-NN‍δn1(θ,fn-fˇn)eiπnx.
Reiteration of this transformation up to p times leads to the following expansion of the error:
(15)rN(f,x)=(e-iπNx-eiπ(N+1)x)×∑k=1p‍θkδNk-1(θ,fˇn)∏s=1k(1+θseiπx)(1+θse-iπx)+(eiπNx-e-iπ(N+1)x)×∑k=1p‍θkδ-Nk-1(θ,fˇn)∏s=1k(1+θseiπx)(1+θse-iπx)+1∏s=1p(1+θseiπx)(1+θse-iπx)∑|n|=N+1∞‍δnp(θ,fn)eiπnx+1∏s=1p(1+θseiπx)(1+θse-iπx)×∑n=-NN‍δnp(θ,fn-fˇn)eiπnx,
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:
(16)INp(f,x)=IN(f,x)+(e-iπNx-eiπ(N+1)x)×∑k=1p‍θkδNk-1(θ,fˇn)∏s=1k(1+θseiπx)(1+θse-iπx)+(eiπNx-e-iπ(N+1)x)×∑k=1p‍θkδ-Nk-1(θ,fˇn)∏s=1k(1+θseiπx)(1+θse-iπx),
with the error
(17)rNp(f,x)=1∏s=1p(1+θseiπx)(1+θse-iπx)×∑|n|=N+1∞‍δnp(θ,fn)eiπnx+1∏s=1p(1+θseiπx)(1+θse-iπx)×∑n=-NN‍δnp(θ,fn-fˇn)eiπnx.The problem of the determination of parametersθk 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].Letf∈Cq-1[-1,1]. By Ak(f), we denote the jumps of f at the end points of the interval
(18)Ak(f)=f(k)(1)-f(k)(-1),k=0,…,q-1.The polynomial correction method is based on the following representation of the interpolated function:(19)f(x)=∑k=0q-1‍Ak(f)Bk(x)+F(x),
where Bk are 2-periodic Bernoulli polynomials
(20)B0(x)=x2,Bk(x)=∫‍Bk-1(x)dx,∫-11‍Bk(x)dx=0,x∈[-1,1],
with the Fourier coefficients
(21)Bn(k)=(-1)n+12(iπn)k+1,n≠0,B0(k)=0.
Function F is a 2-periodic and relatively smooth function on the real line (F∈Cq-1(R)) with the discrete Fourier coefficients
(22)Fˇn=fˇn-∑k=0q-1‍Ak(f)Bˇn(k).
The approximation of F in (19) by the classical trigonometric interpolation leads to the Krylov-Lanczos (KL-) interpolation
(23)IN,q(f,x)=∑k=0q-1‍Ak(f)Bk(x)+IN(F,x),
and the approximation of F by the rational-trigonometric interpolation leads to the rational-trigonometric-polynomial (rtp-) interpolation
(24)IN,qp(f,x)=∑k=0q-1‍Ak(f)Bk(x)+INp(F,x)
with the errors
(25)rN,q(f,x)=f(x)-IN,q(f,x),(26)rN,qp(f,x)=f(x)-IN,qp(f,x),
respectively.We need the next results for further comparisons. Theorems1 and 2 show the behavior of the KL-interpolation in the regions away from the singularities (x=±1).Denote(27)ϕm=∑s=-∞∞‍(-1)s(2s+1)m.Theorem 1 (see [1]).
Letq≥2 be even, f∈Cq+1[-1,1], and f(q+1)∈AC[-1,1]. Then, the following estimate holds for |x|<1,
(28)rN,q(f,x)=φN,q(f,x)Nq+1+o(N-q-1),N→∞,
where
(29)φN,q(f,x)=Aq(f)(-1)N+q/22πq+1sin(π(2N+1)x/2)cos(πx/2)ϕq+1.Theorem 2 (see [1]).
Letq≥1 be odd, f∈Cq+2[-1,1], and f(q+2)∈AC[-1,1]. Then, the following estimate holds for |x|<1,
(30)rN,q(f,x)=φN,q(f,x)Nq+2+o(N-q-2),N→∞,
where
(31)φN,q(f,x)=Aq(f)(-1)N+(q+1)/2+1(q+1)4πq+1×sin(πx/2)sin(π(2N+1)x/2)cos2(πx/2)ϕq+2+Aq+1(f)(-1)N+(q+1)/22πq+2sin(π(2N+1)x/2)cos(πx/2)ϕq+2.The aim of this paper is the derivation of analogs of Theorems1 and 2 for the rtp-interpolations.The determination of parametersθk is crucial for the realization of the rtp-interpolation. General method leads to theFourier-Padé interpolation (see [16]).Here, we consider a smooth functionf on [-1,1] and take
(32)θk=1-τkN,k=1,…,p,
where the new parameters τk 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 τk from the condition
(33)limN→∞Nq+1/2∥rN,qp(f,x)∥L2→minimum.
Paper [18] showed the solution of that problem for p=1 and 1≤q≤6.Another approach for the determination of parametersτk was described in [17], where τk were the roots of the associated Laguerre polynomial Lpq(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 θk chosen as in (32).Theorem 3 (see [15]).
Letf∈Cq+2p+1[-1,1] and f(q+2p+1)∈AC[-1,1] for some q,p≥1. Let parameters θk be chosen as in (32). Then, the following estimate holds as N→∞ and |x|<1(34)rN,qp(f,x)=O(N-q-2p-1).The main purpose of this paper is the derivation of the exact estimates of therN,qp(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 jumpsAk(f) are known and that interpolated function is smooth on [-1,1].


## 4. Pointwise Convergence of the RTP-Interpolation

Letθk be defined as in (32) and by γk(τ) denote the coefficients of the polynomial
(35)∏s=1p‍(1+τsx)=∑s=0p‍γs(τ)xs.Also denote(36)ψm,p=∑s=0p‍(-1)sγs(τ)∑k=0p‍γk(τ)(2p-k-s+m)!ϕ2p-k-s+m+1.Theorems4 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.
Letq≥2 be even and f∈Cq+2p+1[-1,1] with f(q+2p+1)∈AC[-1,1] for some p≥1. Let parameters θk be chosen as in (32). Then, the following estimate holds for |x|<1,
(37)rN,qp(f,x)=φN,q,p(f,x)N2p+q+1+o(N-2p-q-1),N→∞,
where
(38)φN,q,p(f,x)=Aq(f)(-1)N+p+q/222p+1πq+1q!sin(π(2N+1)x/2)cos2p+1(πx/2)ψq,p.Proof.
Expansion (19) and definition of interpolation IN,qp(f,x) show that
(39)rN,qp(f,x)=rNp(F,x).
The application of transformation (14) to (17) with θp+1=1 implies
(40)rN,qp(f,x)=δNp(θ,Fˇn)c(x)(e-iπNx-eiπ(N+1)x)+δ-Np(θ,Fˇn)c(x)(eiπNx-e-iπ(N+1)x)+1c(x)∑n=-NN‍δn1(δnp(θ,Fn-Fˇn))eiπnx+1c(x)∑|n|=N+1∞‍δn1(δnp(θ,Fn))eiπnx,
where, by δns(yn), we denoted δns(θ,yn) with θk=1, k=1,…,s and
(41)c(x)=(1+eiπx)(1+e-iπx)∏s=1p‍(1+θseiπx)(1+θse-iπx)=4cos2πx2∏s=1p‍(1+2θscosπx+θs2).
First, we estimate the last term in the right-hand side of (40). We need to estimate δn1(δnp(θ,Fn)) for |n|>N as N→∞. In view of the smoothness of F, we get from expansion (19) by means of integration by parts
(42)Fn=∑m=qq+2p+1‍Am(f)Bn(m)+12(iπn)2p+q+2×∫-11‍f(2p+q+2)(x)e-iπnxdx=∑m=qq+2p+1‍Am(f)Bn(m)+o(n-2p-q-2),
and consequently
(43)δn1(δnp(θ,Fn))=∑m=qq+2p+1‍Am(f)δn1(δnp(θ,Bn(m)))+o(n-2p-q-2).
According to Lemma A.1,
(44)δn1(δnp(θ,Bn(m)))=1N2pO(n-m-3),
and, therefore,
(45)δn1(δnp(θ,Fn))=1N2po(n-q-2).
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 δn1(δnp(θ,Fn-Fˇn)) for |n|≤N as N→∞. Observing that
(46)Fˇn-Fn=∑s≠0‍Fn+s(2N+1),
and, applying (42), we obtain
(47)Fˇn-Fn=∑m=qq+2p+1‍Am(f)(∑s≠0‍Bn+s(2N+1))+1N2p+q+2×∑s≠0‍εn+s(2N+1)(n/(2N+1)+s)q+2p+2=∑m=qq+2p+1‍Am(f)(Bˇn(m)-Bn(m))+o(N-2p-q-2),|n|≤N,N→∞,
where, by εn, we denoted
(48)εn=12(iπ)2p+q+2∫-11‍f(2p+q+2)(x)e-iπnxdx
and took into account that εn=o(1) as n→∞. Now, it follows that
(49)δn1(δnp(θ,Fˇn-Fn))=∑m=qq+2p+1‍Am(f)δn1(δnp(θ,Bˇn(m)-Bn(m)))+o(N-2p-q-2).
According to Lemma A.2,
(50)δn1(δnp(θ,Bˇn(m)-Bn(m)))=O(N-2p-q-3),
and, therefore,
(51)δn1(δnp(θ,Fˇn-Fn))=o(N-2p-q-2).
Hence, the third term in the right-hand side of (40) is also o(N-q-2p-1) as N→∞.
Finally, we get(52)rN,qp(f,x)=δNp(θ,Fˇn)c(x)(e-iπNx-eiπ(N+1)x)+δ-Np(θ,Fˇn)c(x)(eiπNx-e-iπ(N+1)x)+o(N-q-2p-1),
and we need to estimate δ±Np(θ,Fˇn). Similarly, as above,
(53)δ±Np(θ,Fˇn)=∑m=qq+2p+1‍Am(f)δ±Np(θ,Bˇn(m))+o(N-2p-q-2).
In view of Lemmas A.3 and A.4,
(54)δ±Np(θ,Fˇn)=Aq(f)δ±Np(θ,Bˇn(q))+o(N-2p-q-2)=±Aq(f)(-1)N+p+12(iπ)q+1N2p+q+1q!ψq,p+O(N-2p-q-2).
Substituting this into (52), we get the required estimate as θk→1 and
(55)limN→∞c(x)=22p+2cos2p+2πx2.We prove similar result for odd values ofq.Theorem 5.
Letq be odd, q≥1, and f∈Cq+2p+2[-1,1] with f(q+2p+2)∈AC[-1,1] for some p≥1. Let parameters θk be chosen as in (32). Then, the following estimate holds for |x|<1,
(56)rN,qp(f,x)=φN,q,p(f,x)N2p+q+2+o(N-2p-q-2),N→∞,
where
(57)φN,q,p(f,x)=Aq(f)(-1)N+p+(q+1)/2+122p+2πq+1q!×sin(πx/2)sin(π(2N+1)x/2)cos2p+2(πx/2)ψq+1,p+Aq+1(f)(-1)N+p+(q+1)/222p+1πq+2(q+1)!×sin(π(2N+1)x/2)cos2p+1(πx/2)ψq+1,p.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 θp+1=θp+2=1 implies that(58)rN,qp(f,x)=δNp(θ,Fˇn)c(x)(e-iπNx-eiπ(N+1)x)+δ-Np(θ,Fˇn)c(x)(eiπNx-e-iπ(N+1)x)+δN1(δnp(θ,Fˇn))d(x)(e-iπNx-eiπ(N+1)x)+δ-N1(δnp(θ,Fˇn))d(x)(eiπNx-e-iπ(N+1)x)+1d(x)∑n=-NN‍δn2(δnp(θ,Fn-Fˇn))eiπnx+1d(x)∑|n|=N+1∞‍δn2(δnp(θ,Fn))eiπnx,
where, by δns(yn), we denoted δns(θ,yn) with θk=1, k=1,…,s, and
(59)c(x)=4cos2πx2∏s=1p‍(1+2θscosπx+θs2),d(x)=(1+eiπx)2(1+e-iπx)2∏s=1p‍(1+θseiπx)(1+θse-iπx)=16cos4πx2∏s=1p‍(1+2θscosπx+θs2).
We have(60)Fn=∑m=qq+2p+2‍Am(f)Bn(m)+o(n-2p-q-3),δn2(δnp(θ,Fn))=∑m=qq+2p+2‍Am(f)δn2(δnp(θ,Bn(m)))+o(n-2p-q-3).
According to Lemma A.1,
(61)δn2(δnp(θ,Fn))=1N2po(n-q-3),
and the last term in the right-hand side of (58) is o(N-q-2p-2) as N→∞.
Similarly,(62)δn2(δnp(θ,Fˇn-Fn))=∑m=qq+2p+2‍Am(f)δn2(δnp(θ,Bˇn(m)-Bn(m)))+o(N-2p-q-3),
and, according to Lemma A.2,
(63)δn2(δnp(θ,Fˇn-Fn))=o(N-2p-q-3),N→∞.
Hence, the fifth term in the right-hand side of (58) is o(N-q-2p-2) as N→∞.
Then,(64)δ±Nw(δnp(θ,Fˇn))=∑m=qq+2p+2‍Am(f)δ±Nw(δnp(θ,Bˇn(m)))+o(N-2p-q-3).
According to Lemma A.4, when w=1 in (64) and q is odd, we get
(65)δ±N1(δnp(θ,Fˇn))=Aq(f)δ±N1(δnp(θ,Bˇn(q)))+o(N-2p-q-3)=o(N-2p-q-3),
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
(66)rN,qp(f,x)=δNp(θ,Fˇn)c(x)(e-iπNx-eiπ(N+1)x)+δ-Np(θ,Fˇn)c(x)(eiπNx-e-iπ(N+1)x)+o(N-q-2p-2).
Taking w=0 in (64), we write
(67)δ±Np(θ,Fˇn)=∑m=qq+2p+2‍Am(f)δ±Np(θ,Bˇn(m))+o(N-2p-q-3)=Aq(f)δ±Np(θ,Bˇn(q))+Aq+1(f)δ±Np(θ,Bˇn(q+1))+o(N-2p-q-3).
Now, the application of Lemmas A.2 and A.3 completes the proof.Note that Theorems4 and 5 are valid also for p=0. In that case, the rtp-interpolation coincides with the KL-interpolation as IN,q0(f,x)≡IN,q(f,x) and consequently rN,q0(f,x)≡rN,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 Theorems4 and 5. Theorem 4 states that on the interval |x|<1, the rate of convergence of IN,qp(f,x) is O(N-q-2p-1) for even values of q. According to Theorem 5, the rate of convergence of IN,qp(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 Aq(f), the estimate of Theorem 5 depends on both Aq(f) and Aq+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 parametersτk in (32) are the roots of the associated Laguerre polynomial Lpq(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 givenq 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(N2p).Now, we consider functions with finite smoothness and suppose that all needed jumps are exactly available. Estimates proved above show (see factorsAq(f) and Aq+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 parametersτk are the roots of the associated Laguerre polynomials Lpq(x).Letf∈CM+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
(68)f(x)=sin(ax-1),
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-1maxx∈[-0.5,0.5]|φN,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 ofN-q-2p-1maxx∈[-0.5,0.5]|φN,q,p(f,x)| (see (38)) for N=1024, q+2p=8 and for function (68). Parameters τk are the roots of the associated Laguerre polynomials Lpq(x).
p
=
3
p
=
2
p
=
1
p
=
0
q
=
2
q
=
4
q
=
6
q
=
8
a
=
1
/
10
1.5
·
1
0
-
30
4.6
·
1
0
-
34
1.2
·
1
0
-
37
3.9
·
1
0
-
41
a
=
1
1.2
·
1
0
-
27
3.9
·
1
0
-
29
1.1
·
1
0
-
30
3.3
·
1
0
-
32
a
=
10
8.0
·
1
0
-
26
2.5
·
1
0
-
25
6.8
·
1
0
-
25
2.2
·
1
0
-
24
a
=
30
1.3
·
1
0
-
24
3.7
·
1
0
-
23
9.0
·
1
0
-
22
2.6
·
1
0
-
22
a
=
50
9.7
·
1
0
-
25
7.6
·
1
0
-
23
5.1
·
1
0
-
21
4.1
·
1
0
-
19Table1 shows that for a=1/10 and a=1 utilization of all available jumps is reasonable and the KL-interpolation IN,8(f,x) has the best accuracy. For a=10,30, and 50, the rtp-interpolation IN,23(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 Table1 are obtained based on asymptotic estimates and we must compare these results to actual errors maxx∈[-0.5,0.5]|rN,qp(f,x)| to see how the asymptotic errors coincide with actual errors. In Table 2, the values of the actual errors maxx∈[-0.5,0.5]|rN,qp(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 ofmaxx∈[-0.5,0.5]|rN,qp(f,x)| (see (26)) for N=1024, q+2p=8, and for function (68). Parameters τk are the roots of the associated Laguerre polynomials Lpq(x).
p
=
3
p
=
2
p
=
1
p
=
0
q
=
2
q
=
4
q
=
6
q
=
8
a
=
1
/
10
1.2
·
1
0
-
30
3.9
·
1
0
-
34
1.2
·
1
0
-
37
3.9
·
1
0
-
41
a
=
1
9.8
·
1
0
-
28
3.3
·
1
0
-
29
9.9
·
1
0
-
31
3.3
·
1
0
-
32
a
=
10
6.3
·
1
0
-
26
2.1
·
1
0
-
25
6.4
·
1
0
-
25
2.1
·
1
0
-
24
a
=
30
1.0
·
1
0
-
24
3.1
·
1
0
-
23
8.5
·
1
0
-
22
2.6
·
1
0
-
22
a
=
50
7.8
·
1
0
-
25
6.4
·
1
0
-
23
4.8
·
1
0
-
21
4.0
·
1
0
-
19We have the same situation for odd values ofq. Let f∈CM+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/Nq+2p+2)maxx∈[-0.5,0.5]|φN,q,p(f,x)| and in Table 4 to  calculate the values of maxx∈[-0.5,0.5]|rN,qp(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 IN,13(f,x).Table 3
Values ofN-2p-q-2maxx∈[-0.5,0.5]|φN,q,p(f,x)| (see (57)) for N=1024, q+2p=7 and for function (68). Parameters τk are the roots of the associated Laguerre polynomials Lpq(x).
p
=
3
p
=
2
p
=
1
p
=
0
q
=
1
q
=
3
q
=
5
q
=
7
a
=
1
/
10
4.0
·
1
0
-
28
2.0
·
1
0
-
31
4.8
·
1
0
-
35
7.1
·
1
0
-
39
a
=
1
4.0
·
1
0
-
26
1.9
·
1
0
-
27
4.3
·
1
0
-
29
6.3
·
1
0
-
31
a
=
10
6.6
·
1
0
-
25
2.3
·
1
0
-
24
4.5
·
1
0
-
24
6.0
·
1
0
-
24
a
=
30
8.6
·
1
0
-
24
2.3
·
1
0
-
22
3.7
·
1
0
-
21
4.1
·
1
0
-
20
a
=
50
6.0
·
1
0
-
24
4.3
·
1
0
-
22
1.9
·
1
0
-
20
5.5
·
1
0
-
19Table 4
Values ofmaxx∈[-0.5,0.5]|rN,qp(f,x)| (see (26)) for N=1024, q+2p=7 and for function (68). Parameters τk are the roots of the associated Laguerre polynomials Lpq(x).
p
=
3
p
=
2
p
=
1
p
=
0
q
=
1
q
=
3
q
=
5
q
=
7
a
=
1
/
10
3.8
·
1
0
-
28
2.0
·
1
0
-
31
4.7
·
1
0
-
35
7.0
·
1
0
-
39
a
=
1
3.8
·
1
0
-
26
1.8
·
1
0
-
27
4.2
·
1
0
-
29
6.2
·
1
0
-
31
a
=
10
6.3
·
1
0
-
25
2.2
·
1
0
-
24
4.5
·
1
0
-
24
6.0
·
1
0
-
24
a
=
30
8.1
·
1
0
-
24
2.3
·
1
0
-
22
3.7
·
1
0
-
21
4.1
·
1
0
-
20
a
=
50
5.7
·
1
0
-
24
4.2
·
1
0
-
22
1.8
·
1
0
-
20
5.5
·
1
0
-
19Overall 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 factorsAq(f) and Aq+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 Aq(f) and Aq+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.



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*Source: 315748-2013-03-21.xml*