# On a Pointwise Convergence of Quasi-Periodic-Rational Trigonometric Interpolation

Authors: Arnak Poghosyan; Lusine Poghosyan
Journal: International Journal of Analysis
 (2014)

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## Abstract

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.


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## Body

## 1. Introduction

The quasi-periodic (QP) interpolationIN,m(f,x), m≥0 (m is integer) and x∈[-1,1], interpolates function f on equidistant grid
(1)xk=kN,|k|≤N
and is exact for a quasi-periodic function
(2)eiπnσx,|n|≤N,σ=2N2N+m+1
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 L2(-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.Let(3)Ask(f)=f(k)(1)-(-1)k+sf(k)(-1),k=0,…,q,Φk,m(eiπx)=e(iπ/2)(m-1)x∑r=-∞∞‍(-1)r(m+1)(2r+x)k+1.We denote byRN,m(f,x) the error of the QP interpolation as follows:
(4)RN,m(f,x)=f(x)-IN,m(f,x).Theorem 1 (see [6]).
Letf(q+2m)∈AC[-1,1] for some m≥1, q≥0, and
(5)f(k)(-1)=f(k)(1)=0,k=0,…,q-1.
Then, the following estimate holds for |x|<1 as N→∞:(6)RN,m(f,x)=Cq,m(f)(-1)NNq+m+1×[sin(π(N+1)σx)∑k=0[m/2]‍(m-kk)×(-1)k22k+1cos⁡2k+2(πx/2)-sin(πNσx)×∑k=0[m/2]-1‍(m-k-2k)×(-1)k22k+3cos⁡2k+4(πx/2)∑k=0[m/2](m-k-2k)‍]+o(N-q-m-1),
where
(7)Cq,m(f)=∑k=0q‍Akq(f)(m+1)q-k2q-k+1ik-1πk-m+1(q-k)!Φk,m(m)(-1).Theorem 2 (see [6]).
Letf(q+1)∈AC[-1,1] for some q≥0 and
(8)f(k)(-1)=f(k)(1)=0,k=0,…,q-1.
Then, the following estimate holds for |x|<1 as N→∞:
(9)RN,0(f,x)=A0q(f)(-1)N2q+1Nq+1sin⁡(πNx)cos⁡(πx/2)×∑k=0[q/2]‍(-1)k22k(q-2k)!π2k+1∑s=-∞∞‍(-1)s(2s+1)2k+1+o(N-q-1).In the current paper, we consider convergence acceleration of the QP interpolation by rational corrections in terms ofeiπσx 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:(10)IN,m(f,x)=∑n=-NN‍Fn,meiπnσx,m∈Z,m≥0,
where
(11)Fn,m=f̌n,m-∑ℓ=1m‍θn,ℓf̌ℓ+N,m,f̌n,m=12N+m+1∑k=-NN‍f(kN)e-2iπnk/(2N+m+1),θn,ℓ=e2iπ(ℓ+N-n)(N+m)/(2N+m+1)∑s=1m‍vℓ,s-1e2iπn(s-1)/(2N+m+1).
Here, vℓ,s-1 are the elements of the inverse of the Vandermonde matrix (vs,ℓ) as
(12)vs,ℓ=αℓs-1,αℓ=e2iπ(ℓ+N)/(2N+m+1),
and have the following explicit form [8]:
(13)vℓ,s-1=-1αℓs∏i=1,i≠ℓm(αℓ-αi)∑j=0s-1‍βjαℓj,ℓ,s=1,…,m,
where βj are the coefficients of the following polynomial:
(14)∏j=1m‍(x-αj)=∑j=0m‍βjxj.Taking into account thatαs-αi=O(1/N), from (13), we get
(15)vℓ,s-1=O(Nm-1),N⟶∞,(16)θn,ℓ=O(Nm-1),N⟶∞.


## 2. Quasi-Periodic-Rational Interpolation

In this section, we consider convergence acceleration of the QP interpolation by rational trigonometric corrections which leads to the QPR interpolation.Consider a vectorμ={μk}k=1p. By Δnk(μ,cn), we denote generalized finite differences defined by the following recurrent relations:
(17)Δn0(μ,cn)=cn,Δnk(μ,cn)=Δnk-1(μ,cn)+μkΔn-1k-1(μ,cn),k≥1,
for some sequence cn. When μ≡1, we put
(18)Δnk(μ,cn)=Δnk(cn).
It is easy to verify that
(19)Δn1(μ,cn)=cn+μ1cn-1,Δn2(μ,cn)=Δn1(μ,cn)+μ2Δn-11(μ,cn)Δn2(μ,cn)=cn+(μ1+μ2)cn-1+μ1μ2cn-2,Δn3(μ,cn)=cn+(μ1+μ2+μ3)cn-1Δn3(μ,cn)=+(μ1μ2+μ1μ3+μ2μ3)cn-2+μ1μ2μ3cn-3.
In general, we can prove by the mathematical induction that
(20)Δnp(μ,cn)=∑s=0p‍γs(μ)cn-s,
where γs(μ) are the coefficients of the following polynomial:
(21)∑s=0p‍γs(μ)xs=∏s=1p‍(1+μsx).Consider the following vectors:λ={λk}|k|=1p, λ+={λk}k=1p, and λ-={λ-k}k=1p. By δnk(λ,cn), we denote modified finite differences defined by the following recurrent relations:
(22)δn0(λ,cn)=cn,δnk(λ,cn)=δnk-1(λ,cn)+λ-kδn-1k-1(λ,cn)+λk(δn+1k-1(λ,cn)+λ-kδnk-1(λ,cn)),k≥1,
for some sequence cn. When λ≡1, we put
(23)δnk(λ,cn)=δnk(cn).
Similar to (20), we can show that
(24)δnp(λ,cn)=∑k=0p‍γk(λ+)∑s=0p‍γs(λ-)cn+k-s,
where
(25)∏k=1p‍(1+λkx)=∑k=0p‍γk(λ+)xk,∏k=1p‍(1+λ-kx)=∑k=0p‍γk(λ-)xk.It is easy to verify that(26)δnk(cn)=Δn+k2k(cn),(27)Δnk(cn)=∑s=0k‍(ks)cn-s,(28)δnk(cn)=∑s=02k‍(2ks)cn+k-s.We assume thatf∈Cα[-1,1] for some α≥1 and we denote
(29)f*(x)={fleft(x),x∈[-1,-σ),f(xσ),x∈[-σ,σ],fright(x),x∈(σ,1],
where
(30)fleft(x)=∑j=0α‍f(j)(-1)j!(xσ+1)j,fright(x)=∑j=0α‍f(j)(1)j!(xσ-1)j.According to definition off*, we can write
(31)f*(x)=∑n=-∞∞‍fn*eiπnx,x∈(-1,1).
Hence,
(32)f(x)=∑n=-∞∞‍fn*eiπnσx,x∈[-1,1].
Therefore,
(33)RN,m(f,x)=∑n=-NN‍(fn*-Fn,m)eiπnσx+∑|n|>N‍fn*eiπnσx.The following transformation is easy to verify (see details in [9] for similar transformation):
(34)RN,m(f,x)=λ-1F-N-1,me-iπσNx-FN,meiπσ(N+1)x(1+λ-1eiπσx)(1+λ1e-iπσx)+λ1FN+1,meiπσNx-F-N,me-iπσ(N+1)x(1+λ-1eiπσx)(1+λ1e-iπσx)+1(1+λ-1eiπσx)(1+λ1e-iπσx)×∑n=-NN‍δn1(λ,fn*-Fn,m)eiπnσx+1(1+λ-1eiπσx)(1+λ1e-iπσx)×∑|n|>N‍δn1(λ,fn*)eiπnσx.
Reiteration of it up to p times leads to the following expansion of the error:
(35)RN,m(f,x)=∑k=1p‍λ-kδ-N-1k-1(λ,Fn,m)e-iπσNx-δNk-1(λ,Fn,m)eiπσ(N+1)x∏s=1k(1+λ-seiπσx)(1+λse-iπσx)+∑k=1p‍λkδN+1k-1(λ,Fn,m)eiπσNx-δ-Nk-1(λ,Fn,m)e-iπσ(N+1)x∏s=1k(1+λ-seiπσx)(1+λse-iπσx)+1∏s=1p(1+λ-seiπσx)(1+λse-iπσx)×∑|n|≤N‍δnp(λ,fn*-Fn,m)eiπnσx+1∏s=1p(1+λ-seiπσx)(1+λse-iπσx)×∑|n|>N‍δnp(λ,fn*)eiπnσx,
where the first two terms can be assumed as corrections of the error. This observation leads to the following QPR interpolation:
(36)IN,mp(f,x)=IN,m(f,x)+∑k=1p‍λ-kδ-N-1k-1(λ,Fn,m)e-iπσNx-δNk-1(λ,Fn,m)eiπσ(N+1)x∏s=1k(1+λ-seiπσx)(1+λse-iπσx)+∑k=1p‍λkδN+1k-1(λ,Fn,m)eiπσNx-δ-Nk-1(λ,Fn,m)e-iπσ(N+1)x∏s=1k(1+λ-seiπσx)(1+λse-iπσx)
with the error
(37)RN,mp(f,x)=1∏s=1p(1+λ-seiπσx)(1+λse-iπσx)×∑|n|≤N‍δnp(λ,fn*-Fn,m)eiπσnx+1∏s=1p(1+λ-seiπσx)(1+λse-iπσx)×∑|n|>N‍δnp(λ,fn*)eiπσnx.The QPR interpolation is undefined until parametersλk are unknown. Hence, determination of these parameters is a crucial problem for realization of the QPR interpolation. First, we assume that
(38)λ-k=λk=1-τkN,k=1,…,p,
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 Fourier-Pade interpolation which leads to quasi-periodic Fourier-Pade interpolation.


## 3. Convergence Analysis

Letλk be chosen as in (38) and let γk(τ) be the coefficients of the following polynomial:
(39)∏s=1p‍(1+τsx)=∑k=0p‍γk(τ)xk,
where τ={τ1,…,τp}.Let us modify (20) in view of (38). For p=1, we write
(40)Δn1(λ+,cn)=cn+(1-τ1N)cn-1=Δn1(cn)-τ1Ncn-1.
For p=2, we have
(41)Δn2(λ+,cn)=Δn1(λ+,cn)+(1-τ2N)Δn-11(λ+,cn)=Δn1(cn)-τ1Ncn-1+(1-τ2N)(Δn-11(cn)-τ1Ncn-2)=Δn2(cn)-τ1+τ2NΔn-11(cn)+τ1τ2N2Δn-20(cn).
In general, we can prove by the mathematical induction the following expansion [10]:
(42)Δnp(λ+,cn)=∑s=0p‍γs(λ+)cn-s=∑s=0p‍(-1)sγs(τ)NsΔn-sp-s(cn).Now, let us modify (24) in view of (38). According to (20) and (42), we get (note that λ+=λ-)
(43)δnp(λ,cn)=∑k=0p‍γk(λ+)∑s=0p‍γs(λ-)cn+k-s=∑s=0p‍(-1)sγs(τ)Ns∑k=0p‍γk(λ+)Δn+k-sp-s(cn).
Similar to (42), we can show that
(44)∑k=0p‍γk(λ+)cn+k=∑k=0p‍(-1)kγk(τ)NkΔn+pp-k(cn).
Then, from (43), we have
(45)δnp(λ,cn)=∑s=0p‍(-1)sγs(τ)Ns∑k=0p‍(-1)kγk(τ)NkΔn+pp-k(Δn-sp-s(cn)).
This leads to the following needed expansion:
(46)δnp(λ,cn)=∑s=0p‍(-1)sγs(τ)Ns∑k=0p‍(-1)kγk(τ)NkΔn+p-s2p-k-s(cn)
as Δn+pp-k(Δn-sp-s(cn))=Δn+p-s2p-k-s(cn).Then,(47)δnw(δnp(λ,cn))=∑s=0p‍(-1)sγs(τ)Ns∑k=0p‍(-1)kγk(τ)NkΔn+w2w(Δn+p-s2p-k-s(cn)).
Taking into account that
(48)Δn+w2w(Δn+p-s2p-k-s(cn))=Δn+p-s+w2w+2p-k-s(cn),
we find that
(49)δnw(δnp(λ,cn))=∑s=0p‍(-1)sγs(τ)Ns∑k=0p‍(-1)kγk(τ)NkΔn+p-s+w2w+2p-k-s(cn).
We will frequently use the latest formula.We denote byfn the nth Fourier coefficient of f as
(50)fn=12∫-11‍f(x)e-iπnxdx.Let(51)Bn(j)=(-1)n+12(iπn)j+1,n≠0,B0(j)=0,ψp,m,j±(τ)=∑s=0p‍(-1)sγs(τ)∑k=0p‍γk(τ)(2p-k-s+j)!×∑r=-∞∞‍(-1)r(m+1)(2r±1)2p-k-s+j+1.First, we prove some lemmas.Lemma 3.
Letf(q+2p+m)∈AC[-1,1] for some q≥0, p,m≥1, and
(52)f(k)(-1)=f(k)(1)=0,k=0,…,q-1.
Let parameters λk be chosen as in (38). Then,
(53)δnw(δnp(λ,fn*))=O(n-2w-1N-2p-q)+o(n-q-2p-m-1),hhhhhhhhhhhhhhhhhhhhhhhhh|n|>N,N⟶∞.Proof.
We have (see details in [5])
(54)fn*=∑j=qq+2p+m‍12jNj∑k=0j‍Akj(f)(m+1)j-k(2N+m+1)k(j-k)!Bn(k)+o(n-q-2p-m-1),
and, consequently,
(55)δnw(δnp(λ,fn*))=∑j=qq+2p+m‍12jNj×∑k=0j‍Akj(f)(m+1)j-k(2N+m+1)k(j-k)!×δnw(δnp(λ,Bn(k)))+o(n-q-2p-m-1).
Taking into account (49) and estimating Δnp(Bn(k))=O(n-k-1-p) (see details in [9]), we get
(56)δnw(δnp(λ,Bn(k)))=O(n-2w-k-1N-2p)
which completes the proof.Lemma 4.
Letλk be chosen as in (38), and let β∈ℝ be a constant. Then, the following estimate holds for n∈ℤ as N→∞:
(57)δnw(δnp(λ,(-1)neiπβn/(2N+m+1)))=(πβ2)2w(-1)neiπβn/(2N+m+1)N2w+2php(β,τ)+O(N-2w-2p-1),
where
(58)hp(β,τ)=(πβ2)2p∑s=0p‍γs(τ)∑k=0p‍(-1)kγk(τ)(2iπβ)k+s.Proof.
From (49), we have
(59)δnw(δnp(λ,(-1)neiπβn/(2N+m+1)))=∑s=0p‍(-1)sγs(τ)Ns∑k=0p‍(-1)kγk(τ)Nk×Δn+p-s+w2w+2p-k-s((-1)neiπβn/(2N+m+1)).
In view of (27), we write
(60)Δnp((-1)neiπβn/(2N+m+1))=(-1)n∑k=0p‍(pk)(-1)keiπβ(n-k)/(2N+m+1)=(-1)neiπβn/(2N+m+1)∑k=0p‍(pk)(-1)k×∑t=0∞‍(iπβ)t(-1)tktt!(2N+m+1)t=(-1)neiπβn/(2N+m+1)∑t=0∞‍(-1)t(iπβ)tt!(2N+m+1)tωp,t,
where
(61)ωp,t=∑s=0p‍(ps)(-1)sst.
Taking into account that ωp,j=0, 0≤j<p and ωp,p=(-1)pp! (see [10], Lemma 2.1), we get
(62)Δnp((-1)neiπβn/(2N+m+1))=(-1)n(iπβ)p2pNpeiπβn/(2N+m+1)+O(N-p-1).
Then,
(63)Δn+p-s+w2w+2p-k-s((-1)neiπβn/(2N+m+1))=(-1)n+p+s+w(iπβ2N)2w+2p-k-seiπβ(n+p-s+w)/(2N+m+1)+O(N-2w-2p+k+s-1)=(-1)n+p+s+weiπβn/(2N+m+1)(iπβ2N)2w+2p-k-s+O(N-2w-2p+k+s-1)
which completes the proof together with (59).Lemma 5.
Letf(q+2p+m)∈AC[-1,1] for some q≥0, p,m≥1, and
(64)f(k)(-1)=f(k)(1)=0,k=0,…,q-1.
Let parameters λk be chosen as in (38). Then, the following estimate holds as N→∞:
(65)δnw(δnp(λ,Fn,m-fn*))=O(N-q-2w-2p-1)+o(N-q-2p-2),hhhhhhhhhhhhhhhhhhhhhhhhhhhhl-N≤n≤N+m.Proof.
First, we estimateFn,m-fn*. We have (see details in [5], at the beginning of the proof of Lemma 5)
(66)Fn,m=∑r=-∞∞‍fn+r(2N+m+1)*-∑ℓ=1m‍θn,ℓ∑r=-∞∞‍fN+ℓ+r(2N+m+1)*,Fn,m-fn*=∑r≠0‍fn+r(2N+m+1)*-∑ℓ=1m‍θn,ℓ∑r=-∞∞‍fN+ℓ+r(2N+m+1)*.
Then, in view of (54), we get
(67)∑r≠0‍fn+r(2N+m+1)*=∑j=qq+2p+m‍12jNj×∑k=0j‍Akj(f)(m+1)j-k(2N+m+1)k(j-k)!×∑r≠0‍Bn+r(2N+m+1)(k)+o(N-q-2p-m-1),∑ℓ=1mθn,ℓ∑r=-∞∞‍fN+ℓ+r(2N+m+1)*=(-1)n+12N+m+1×∑j=qq+2p+m‍1Nj∑k=0j‍Akj(f)(m+1)j-k2j-k(iπ)k+1(j-k)!×e-iπ(m-1)n/(2N+m+1)∑ℓ=1m‍Φk,m(e2iπ(N+ℓ)/(2N+m+1))×∑s=1m‍vℓ,s-1e2ıπn(s-1)/(2N+m+1)+o(N-q-2p-2),
where we used estimate (16).
According to the Taylor expansion(68)Φk,m(e2ıπ(N+ℓ)/(2N+m+1))=∑t=02p+m‍1t!Φk,m(t)(-1)(e2iπ(N+ℓ)/(2N+m+1)+1)t+O(N-2p-m-1)
and relations
(69)∑ℓ=1m‍(e2iπ(N+ℓ)/(2N+m+1)+1)τ∑s=1m‍vℓ,s-1e2ıπn(s-1)/(2N+m+1)=(e2iπn/(2N+m+1)+1)τ,τ=0,…,m-1,
we derive
(70)Fn,m-fn*=∑j=qq+2p+m‍12jNj×∑k=0j‍Akj(f)(m+1)j-k(2N+m+1)k(j-k)!×∑r≠0‍Bn+r(2N+m+1)(k)+(-1)ne-iπ(m-1)n/(2N+m+1)2N+m+1×∑j=qq+2p+m‍1Nj∑k=0j‍Akj(f)(m+1)j-k2j-k(iπ)k+1(j-k)!×[∑t=0m-1‍Φk,m(t)(-1)t!(e2iπn/(2N+m+1)+1)t+∑t=mm+2p‍Φk,m(t)(-1)t!×∑ℓ=1m‍(e2iπ(N+ℓ)/(2N+m+1)+1)t×∑s=1m‍vℓ,s-1e2ıπn(s-1)/(2N+m+1)∑t=0m-1‍Φk,m(t)(-1)t!]+o(N-q-2p-2).
This completes the proof in view of (49), Lemma 4, and the following estimate [9]:
(71)δnw(δnp(λ,∑r≠0‍Bn+r(2N+m+1)(k)))=O(N-2w-2p-k-1).
We used also the fact that e2iπ(N+ℓ)/(2N+m+1)+1=O(1/N).Lemma 6.
Letf(q+2p+m)∈AC[-1,1] for some q≥0, p,m≥1, and
(72)f(k)(-1)=f(k)(1)=0,k=0,…,q-1.
Let parameters λk be chosen as in (38). Then,
(73)δ±Np(λ,Fn,m)=(-1)N+12q+1Nq+2p+1×∑k=0q‍Akq(f)(m+1)q-k2k(q-k)!(iπ)k+1×[∑μ=0t‍(tμ)(-1)pψp,m,k±(τ)k!-∑t=0m-1‍Φk,m(t)(-1)t!×∑μ=0t‍(tμ)e±iπN(2μ-m+1)/(2N+m+1)×hp(2μ-m+1,τ)∑μ=0t‍(tμ)]+O(N-q-2p-2),(74)δ±(N+1)p(λ,Fn,m)=-δ±Np(λ,Fn,m)+O(N-q-2p-2),(75)δ-Np(λ,Fn,m)=(-1)m+1δNp(λ,Fn,m)+O(N-q-2p-2).Proof.
We proceed as in the proof of Lemma5 and derive
(76)δ±Np(λ,Fn,m)=∑j=qq+2p+m‍12jNj×∑k=0j‍Akj(f)(2N+m+1)k(j-k)!(m+1)k-j×δ±Np(λ,∑r=-∞∞‍Bn+r(2N+m+1)(k))+12N+m+1∑j=qq+2p+m‍1Nj×∑k=0j‍Akj(f)(m+1)j-k2j-k(iπ)k+1(j-k)!×[∑t=0m-1‍Φk,m(t)(-1)t!∑μ=0t‍(tμ)×δ±Np(λ,(-1)neiπn(2μ-m+1)/(2N+m+1))+∑t=mm+2p‍Φk,m(t)(-1)t!×∑ℓ=1m‍(e2iπ(N+ℓ)/(2N+m+1)+1)t×∑s=1m‍vℓ,s-1×δ±Np(λ,(-1)neiπn(2s-1-m)/(2N+m+1))∑t=0m-1‍Φk,m(t)(-1)t!∑μ=0t‍(tμ)δ±Np]+o(N-q-2p-2).
This completes the proof in view of Lemma 4 and the following estimate:
(77)δ±Nw(δnp(λ,∑r=-∞∞‍Bn+r(2N+m+1)(j)))=(-1)N+p+w+12(iπN)j+1N2w+2pj!ψp,m,j+2w±(τ)+O(N-2w-2p-j-2).
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.Theorem 7.
Letf(q+2p+m)∈AC[-1,1] for some q≥0, p,m≥1, and
(78)f(k)(-1)=f(k)(1)=0,k=0,…,q-1.
Let parameters λk be chosen as in (38). Then, the following estimate holds for |x|<1:
(79)RN,mp(f,x)=(-1)N(2N)q+2p+1sin(πσ(N+(1/2))x-(πm/2))cos⁡2p+1(πx/2)×∑k=0q‍Akq(f)(m+1)q-k2k(q-k)!ik-mπk+1×(∑μ=0t‍(tμ)(-1)pψp,m,k+(τ)k!-∑t=0m-1‍Φk,m(t)(-1)t!×∑μ=0t‍(tμ)i2μ-m+1hp(2μ-m+1,τ))+o(N-q-2p-1),N⟶∞.Proof.
We have from (37) by the Abel transformation (see transformation from (33) to (34) with λ≡1)
(80)RN,mp(f,x)=e-iπσNxδ-N-1p(λ,Fn,m)c(x)-eiπσ(N+1)xδNp(λ,Fn,m)c(x)+eiπσNxδN+1p(λ,Fn,m)c(x)-e-iπσ(N+1)xδ-Np(λ,Fn,m)c(x)+1c(x)∑n=-NN‍δn1(δnp(λ,fn*-Fn,m))eiπσnx+1c(x)∑|n|>N‍δn1(δnp(λ,fn*))eiπσnx,
where
(81)c(x)=4cos⁡2(πσx2)∏s=1p‍(1+λ-seiπσx)(1+λse-iπσx).
It is easy to verify that
(82)lim⁡N→∞c(x)=22p+2cos⁡2p+2(πx2).
According to Lemma3,
(83)δn1(δnp(λ,fn*))=O(n-3N-2p-q)+o(n-q-2p-2),hhhhhhhhhhhhhhhhhhhh|n|>N,N⟶∞.
Hence, the last term in the right-hand side of (80) is o(N-q-2p-1). Then, by Lemma 5,
(84)δn1(δnp(λ,Fn,m-fn*))=O(N-q-2p-3)+o(N-q-2p-2),hhhhhhhhhhhhhhhhl-N≤n≤N+m,N⟶∞
and the fifth term is also o(N-q-2p-1).
Therefore,(85)c(x)RN,mp(f,x)=e-iπσNxδ-N-1p(λ,Fn,m)-eiπσ(N+1)xδNp(λ,Fn,m)+eiπσNxδN+1p(λ,Fn,m)-e-iπσ(N+1)xδ-Np(λ,Fn,m)+o(N-q-2p-1).
Taking into account estimates (74) and (75), we get
(86)c(x)RN,mp(f,x)=δNp(λ,Fn,m)×((-1)m(e-iπσNx+e-iπσ(N+1)x)-(eiπσNx+eiπσ(N+1)x))+o(N-q-2p-1)
which concludes the proof in view of Lemma 6.Let us compare the results of Theorems1 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 well-known 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.Note that form=1 the second term in the brackets of estimate (79) vanishes and also ψp,1,0+(1,τ)=0. Hence,
(87)RN,1p(f,x)=o(N-2p-1),N⟶∞,q=0.Similarly, the casem=0 can be analyzed. We present the corresponding theorem without the proof which can be performed as the above one.Theorem 8.
Letf(q+2p+1)∈AC[-1,1] for some q≥0, p≥1, and
(88)f(k)(-1)=f(k)(1)=0,k=0,…,q-1.
Let parameters λk be chosen as in (38). Then, the following estimate holds for |x|<1 as N→∞:
(89)RN,0p(f,x)=(-1)N+p(2N)q+2p+1sin(πNx)cos⁡2p+1(πx/2)×∑k=0q‍Akq(f)2kikπk+1(q-k)!k!ψp,0,k+(τ)+o(N-q-2p-1).Comparison with Theorem2 shows improvement by factor O(N2p) for any p≥1 if f has enough smoothness.


## 4. Parameter Determination in Rational Corrections

Till now, we did not discuss the problem of parametersτk determination as Theorems 7 and 8 are valid for all choices. Now, let us consider some choices with the corresponding numerical results.Let(90)f(x)=(1-x2)4sin(x-1).One choice isτk=k which shows satisfactory numerical results (see Figure 1).The graphs of the absolute errors|R256,mp(f,x)| on interval [-0.7,0.7] for m=2 and p=4 when τk=k (a) and with optimal τk (see (94)) (b) while interpolating (90).
(a)
(b)Another choice is based on the asymptotic estimates of Theorems7 and 8. If it is possible to vanish the following expressions by the choice of parameters τk:
(91)(-1)pψp,m,k+(τ)k!-∑t=0m-1‍Φk,m(t)(-1)t!×∑μ=0t‍(tμ)i2μ-m+1hp(2μ-m+1,τ)=0,k=0,…,q
or
(92)ψp,0,k+(τ)=0,k=0,…,q,
then, Theorems 7 and 8 will be provided with improved estimate
(93)RN,mp(f,x)=o(N-q-2p-1),|x|<1.For example, whenp=q=4 and m=2, we find
(94)τ1=-0.3047909-0.4539705i,τ2=0.9640011-0.1481304i,τ3=6.375385+0.000739i,τ4=12.877189-0.000022i.With this choiceR256,24(f,x)=o(N-13) for (90).Figure1 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 λk immediately along the ideas of the Fourier-Pade interpolation ([11]). This approach is more complex as λk 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. More specifically, parameters λ can be determined from the following system:
(95)δnp(λ,Fn,m)=0,|n|=N-p+1,…,N.
We will refer to this interpolation as quasi-periodic Fourier-Pade interpolation.For example, whenm=2, p=4, and N=256, then, for (90), we get
(96)λ1=0.922+0.010i,λ2=0.952+0.010i,λ3=0.975+0.010i,λ4=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.Figure2 shows the graph of |R256,24(f,x)| for function (90) with λk 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 λk. Theoretical analysis of convergence of the quasi-periodic Fourier-Pade interpolation will be carried out elsewhere.Figure 2
The graph of|R256,mp(f,x)| on interval [-0.7,0.7] for m=2 and p=4 while interpolating (90) by the quasi-periodic Fourier-Pade interpolation (see (96)).



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