# Optimal Rational Approximations by the Modified Fourier Basis

Authors: Arnak V. Poghosyan; Tigran K. Bakaryan
Journal: Abstract and Applied Analysis
 (2018)

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## 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.


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## Body

## 1. Introduction

The modified Fourier basis(1)H=cos⁡πnx:n∈Z+∪sin⁡πn-12x:n∈N,x∈-1,1was originally proposed by Krein [1] and thoroughly investigated in a series of papers [2–10].LetMN(f,x) be the truncated modified Fourier series(2)MNf,x=12f0c+∑n=1Nfnccos⁡πnx+fnssin⁡πn-12x,where(3)fnc=∫-11fxcos⁡πnxdx,fns=∫-11fxsin⁡πn-12xdx.Obviously, for even functions on [-1,1], expansions by the modified Fourier basis coincide with the expansions by the classical Fourier basis(4)Hclass=cos⁡πnx:n∈Z+∪sin⁡πnx:n∈N,x∈-1,1.Moreover, the modified Fourier basis can be derived from the other classical basis H∗ on [0,1](5)H∗=cos⁡πnx:n∈Z+,x∈0,1by 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, 8–10]. We present two theorems for further comparisons.Theorem 1 (see [10]).
Assumef∈C2[-1,1] and f′′∈BV[-1,1]. If x<1, then(6)fx-MNf,x=ON-2,N⟶∞.Otherwise,(7)f±1-MNf,±1=ON-1,N⟶∞.Under some additional requirements, the convergence rate is faster.Theorem 2 (see [2, 10]).
Assumef∈C2q+2(-1,1), f(2q+2)∈BV[-1,1], q≥1, and f obeys the first q derivative conditions:(8)f2r+1±1=0,r=0,…,q-1.If |x|<1, then(9)fx-MNf,x=ON-2q-2,N⟶∞.Otherwise,(10)f±1-MNf,±1=ON-2q-1,N⟶∞.Overall, we see better convergence rates compared to the classical Fourier expansions [11]. This can be explained by faster decay of coefficients fns:(11)fns=On-2,n⟶∞,compared to the classical ones when f is smooth enough but nonperiodic on [-1,1]. Estimate (11) can be explained by a nonperiodicity of the basis functions sin⁡π(n-1/2)x on [-1,1].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 θ={θk}k=1p, p≥1 and, by Δnk(θ,f^), f^=fn denote the following generalized finite differences:(12)Δn0θ,f^=fn,Δnkθ,f^=Δnk-1θ,f^+θkΔn-1k-1θ,f^,k≥1.ByΔnk(f^), we denote the classical finite differences which correspond to generalized differences Δnk(θ,f^) with θ≡1. It is easy to verify that(13)Δnkf^=∑l=0kklfn-l.Let(14)RNf,x=fx-MNf,x=RNcf,x+RNsf,x,where(15)RNcf,x=∑n=N+1∞fnccos⁡πnx,RNsf,x=∑n=N+1∞fnssin⁡πn-12x.Consider two sequences of real numbersθc={θkc}k=1p and θs={θks}k=1p. Let f^s={fns}n=1∞ and f^c={fnc}n=0∞. Let μj(k,θ) be defined by the following identities:(16)∏j=1k1+θjx=∑j=0kμjk,θxj,k=1,…,p.By means of sequential Abel transformations (see details in [6]), we derive the following expansions of errors (15):(17)RNcf,x=-∑k=1pθkcΔNk-1θc,f^c∏r=1k1+2θrccos⁡πx+θrc2∑j=0kμjk,θccos⁡πN+1-jx+RN,pcf,θc,x,RNsf,x=-∑k=1pθksΔNk-1θs,f^s∏r=1k1+2θrscos⁡πx+θrs2∑j=0kμjk,θssin⁡πN+12-jx+RN,psf,θs,x,where(18)RN,pcf,θ,x=12∏k=1p1+θkeiπx∑n=N+1∞Δnpθ,f^ceiπnx+12∏k=1p1+θke-iπx∑n=N+1∞Δnpθ,f^ce-iπnx,(19)RN,psf,θ,x=e-iπx/22i∏k=1p1+θkeiπx∑n=N+1∞Δnpθ,f^seiπnx-eiπx/22i∏k=1p1+θke-iπx∑n=N+1∞Δnpθ,f^se-iπnx.These expansions lead to the following modified-trigonometric-rational (MTR-) approximations:(20)MN,pf,θc,θs,x=MNf,x-∑k=1pθkcΔNk-1θc,f^c∏r=1k1+2θrccos⁡πx+θrc2∑j=0kμjk,θccos⁡πN+1-jx-∑k=1pθksΔNk-1θs,f^s∏r=1k1+2θrscos⁡πx+θrs2∑j=0kμjk,θssin⁡πN+12-jx,with the error(21)RN,pf,θc,θs,x=fx-MN,pf,θc,θs,x=RN,pcf,θc,x+RN,psf,θs,x.A crucial step for realization of the rational approximations is determination of parametersθc and θs. Different approaches are known for solution of this problem (see [12–19]). In general, appropriate determination of these parameters leads to rational approximations with improved accuracy compared to the classical ones in case of smooth f. However, the rational approximations are essentially nonlinear in the sense that(22)MN,pf+g,θc,θs,x≠MN,pf,θc,θs,x+MN,pg,θc,θs,xas for each approximation we need to determine its own θc and θs vectors.In [6], those parameters were determined from the following systems of equations:(23)Δnpθc,f^c=0,n=N,N-1,…,N-p+1,Δnpθs,f^s=0,n=N,N-1,…,N-p+1,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 θc and θs depend on N and systems (23) must be solved for each N.In this paper, assuming thatf is smooth on [-1,1], we consider simpler alternative approach, where θs and θc are determined as follows [14, 16, 19]:(24)θkc=1-τkcN,θks=1-τksN,τkc≠0,τks≠0,k=1,…,p,with τc={τ1c,…,τpc} and τs={τ1s,…,τps} independent of N. Actually, in this approach, we take into consideration only the first two terms of the asymptotic expansions of θk=θk(N) in terms of 1/N. Although parameters θc and θs in (24) depend on N, we need only to determine τc and τs which are independent of N. 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 parametersτc and τs. Second, we determine the optimal values of parameters τc and τs which vanish the main terms and lead to approximations with substantially better pointwise convergence rates. We found that optimal values of parameters τkc and τks, k=1,…,p, are the roots of some polynomials depending on p and q, where q indicates the number of zero derivatives in (8). Moreover, the choice of optimal parameters depends on the parity of p and also on the location of x, whether x<1 or x=±1.For example, whenp is odd and x<1, the roots of the generalized Laguerre polynomial Lp(2q+1)(x) (see Appendix A) could be used for τc and τs. In this case, the convergence rate of the MTR-approximations is O(N-2q-p-(p+1)/2-2) as N→∞. It means better convergence compared to the expansions by the modified Fourier basis with improvement by factor O(Np+p+1/2). When p is odd and x=±1, the roots of the generalized Laguerre polynomial Lp(2q)(x) could be used. In this case, the convergence rate is O(N-2q-p+1/2-1) with the improvement by factor O(Np+1/2). The problem that we encountered is impossibility to get simultaneous optimality on x<1 and at x=±1. One must decide whether to use parameters that provide the minimal error on [-1,1], but with worse accuracy on x<1, or work with optimal parameters on x<1 with worse accuracy at x=±1. We expose similar observations for even values of p.It is important that the values of parametersτc and τs depend only on p and q and are independent of f. It means that if functions f, g, and f+g have enough smoothness and obey the same derivative conditions, the optimal approach leads to linear rational approximations in the sense that(25)MN,pf+g,θc,θs,x=MN,pf,θc,θs,x+MN,pg,θc,θs,xwith the same parameters θc and θs for all included functions.The paper is organized as follows. Section2 presents some preliminary lemmas. Section 3 explores the pointwise convergence of the MTR-approximations away from the endpoints when x<1. Section 4 considers the pointwise convergence of the MTR-approximations when x=±1. 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 parametersθk, k=1,…,p are defined by (see (24))(26)θk=1-τkN,τk≠0,k=1,…,p.Let τ={τ1,…,τp} and let coefficients γk(τ) be defined by the following identity:(27)∏k=1p1+τkx=∑k=0pγkτxk.Forf∈C2q+1[-1,1], q≥0, we put(28)A2k+1f=f2k+11-f2k+1-1-1k,k=0,…,q,B2k+1f=f2k+11+f2k+1-1-1k,k=0,…,q.The following estimates can be simply obtained by means of integration by parts of the integrals in (3).Lemma 3.
Assumef∈C2m+1[-1,1], m≥0, and f(2m+1)∈BV[-1,1]. Then, the following asymptotic expansions are valid:(29)fnc=-1n∑k=0mA2k+1fπn2k+2+on-2m-2,n⟶∞,fns=-1n+1∑k=0mB2k+1fπn-1/22k+2+on-2m-2,n⟶∞.Lemma 4.
Assumef∈C2m+2[-1,1], m≥0, and f(2m+2)∈BV[-1,1]. Then, the following asymptotic expansions are valid:(30)fnc=-1n∑k=0mA2k+1fπn2k+2+on-2m-3,n⟶∞,fns=-1n+1∑k=0mB2k+1fπn-1/22k+2+on-2m-3,n⟶∞.Next lemma unveils the asymptotic expansions ofΔw(Δ^p(θ,f^c)) and Δw(Δ^p(θ,f^s)), where(31)Δ^pθ,f^=Δnpθ,f^n.Lemma 5.
Assumef∈C2q+p+r+1[-1,1], f(2q+p+r+1)∈BV[-1,1], q≥0, r≥0, p≥1, and(32)f2k+1±1=0,k=0,…,q-1.Let θk, k=1,…,p, be defined by (26). Then, the following estimates hold for n>N as N→∞:(33)ΔnwΔ^pθ,f^c=-1n∑k=0pγkτNk∑t=2q+w2q+rt+p-k+1!nt+p-k+2∑s=qt-w/2A2s+1fπ2s+22s+1!βk,s,tw+oN-pn2q+r+2,(34)ΔnwΔ^pθ,f^s=-1n+1∑k=0pγkτNk∑t=2q+w2q+rt+p-k+1!nt+p-k+2∑s=qt-w/2B2s+1fπ2s+22s+1!β~k,s,tw+oN-pn2q+r+2,where(35)βk,s,tw=∑l=wt-2skt-2s-lαw+p-k,l+p-kt-2s-l!p-k+l!,(36)β~k,s,tw=∑l=wt-2sk+12t-2s-lαw+p-k,l+p-kt-2s-l!p-k+l!,with(37)αk,j=∑s=0kks-1ssj,j≥0.Proof.
In view of (12) and (26), we have(38)Δnpθ,f^c=∑k=0p-1kγkτNkΔn-kp-kf^c.Taking into account the fact that Δnw(Δ^p-k(f^c))=Δnw+p-k(f^c) and using (13), we get(39)ΔnwΔ^pθ,f^c=∑k=0p-1kγkτNk∑j=0w+p-kw+p-kjfn-k+jc.Application of Lemma 3, when p+r is even, and Lemma 4, when p + r is odd, leads to the following asymptotic expansion (h=p-k+r):(40)ΔnwΔ^pθ,f^c=-1n∑k=0pγkτNk∑j=0w+p-k-1jw+p-kj∑s=qq+h/2A2s+1fπn2s+211-j+k/n2s+2+oN-pn2q+r+2.Then,(41)ΔnwΔ^pθ,f^c=-1n∑k=0pγkτNk∑j=0w+p-k-1jw+p-kj∑s=qq+h/2A2s+1fπn2s+2∑t=2s+1∞t2s+1k+jt-2s-1nt-2s-1+oN-pn2q+r+2.Finally,(42)ΔnwΔ^pθ,f^c=-1n∑k=0pγkτNk∑t=2q2q+h1nt+2∑s=qt/2A2s+1fπ2s+2t+1t-2s∑l=0t-2st-2slkt-2s-lαw+p-k,l+oN-pn2q+r+2.
It remains to notice thatαw+p-k,l=0 for 0≤l<w+p-k (see [20]). Hence, t-2s≥w+p-k, and s≤[(t-w)/2].


## 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 forx<1 without specifying the selection of parameters τc and τs.Theorem 6.
Assumef∈C2q+p+2[-1,1], f(2q+p+2)∈BV[-1,1], q≥0, p≥1, and(43)f2k+1±1=0,k=0,…,q-1.Let θk, k=1,…,p, be defined by (26). Then, the following estimates hold for |x|<1 as N→∞:(44)RN,pcf,θ,x=A2q+1f-1N+p+1N2q+p+22p+1π2q+22q+1!cos⁡πx/22N-p+1cosp+1⁡πx/2hp,2q+1τ+oN-2q-p-2,(45)RN,psf,θ,x=B2q+1f-1N+pN2q+p+22p+1π2q+22q+1!sin⁡πx/22N-pcosp+1⁡πx/2hp,2q+1τ+oN-2q-p-2,where(46)hp,mτ=∑k=0p-1kγkτm+p-k!.Proof.
We prove only estimate (44). Estimate (45) can be handled similarly.
Taking into account the fact thatθk→1 as N→∞, we estimate only the sums on the right-hand side of (18). By the Abel transformation, we get(47)∑n=N+1∞Δnpθ,f^ce±iπnx=-e±iπN+1x1+e±iπxΔNpθ,f^c-e±iπN+1x1+e±iπx2ΔN1Δ^pθ,f^c+11+e±iπx2∑n=N+1∞Δn2Δ^pθ,f^ce±iπnx.Lemma 5 estimates sequences ΔNp(θ,f^c), ΔN1(Δ^p(θ,f^c)), and Δn2(Δ^p(θ,f^c)) as N→∞ and n≥N+1. It shows that, for r=1 and w=2, we have(48)Δn2Δ^pθ,f^c=oN-pn2q+3,and the third term in the right-hand side of (47) is o(N-p-2q-2). Then, with r=1 and w=1, we have(49)ΔN1Δ^pθ,f^c=ON-p-2q-3,and the second term is O(N-p-2q-3). Finally, using the exact estimate for ΔNp(θ,f^c), we derive(50)ΔNpθ,f^c=A2q+1f-1NNp+2q+2π2q+2∑k=0pγkτ2q+p-k+12q+1αp-k,p-k+ON-2q-p-3,which completes the proof as(51)αp-k,p-k=-1p-kp-k!.Note that Theorem6 is valid also for p=0 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).Theorem6 shows that(52)RN,pf,θc,θs,x=RN,pcf,θc,x+RN,psf,θs,x=ON-2q-p-2if parameters θc and θs are defined by (24). We see improvement in convergence rate by factor O(Np) and this result is obtained without specifying parameters τc and τs.Let us compare the modified Fourier expansions and MTR-approximations for a specific smooth function. Consider the following one:(53)fx=1-x22sin⁡x-1for which(54)f1=f-1=0,f′1=f′-1=0,f′′1=0,f′′-1=-8sin⁡2,f′′′1=24,f′′′-1=24cos⁡2+24sin⁡2.Hence, this function obeys first q=1 derivative conditions (8).Figures1 and 2 show the behaviors of RN (p=0) and RN,p (p=1, 2, 3, 4), respectively, on interval [-0.7,0.7] for N=64 while approximating (53). We used τkc=τks=k, k=1,…,p in the rational approximations.Figure 1
The graph ofRNf,x on [-0.7,0.7] for N=64 while approximating (53) by the modified Fourier expansion (2).Figure 2
The graphs ofRN,pf,θc,θs,x on [-0.7,0.7] for N=64 and different p while approximating (53) by the MTR-approximations with τkc=τks=k, k=1,…,p.According to the results of Theorem6, the bigger the value of p is, the higher the accuracy of the corresponding approximations is. We observe it empirically. We see that max-0.7,0.7⁡RN,p is 3·10-8 for p=0, is 1.6·10-9 for p=1, is 9·10-11 for p=2, is 6.6·10-12 for p=3, and is 5.7·10-13 for p=4.Can we improve the accuracy of the rational approximations by appropriate selection of parametersτs and τc? Further, in this section, we give positive answer to this question and show how the optimal values can be chosen.Estimates of Theorem6 show that improvement can be achieved if parameters are chosen such that τs=τc=τ and(55)hp,2q+1τ=0.By looking into the definition of hp,2q+1(τ), we observe that condition (55) can be achieved, for example, if(56)γkτ=pk2q+1+p!2q+1+p-k!Qrk,where Qr(k) is a polynomial of order r≤p-1(57)Qrk=∑j=0rcjkj,c0=1,with unknown coefficients cj, j=1,…,r. Then, condition (55) follows from the well-known identity(58)∑k=0p-1kpkkj=0,j<p.Further, we determine the values of cj, j=1,…,r, for improved convergence of the rational approximations.Next result is an immediate consequence of those observations and estimates of Theorem6.Theorem 7.
Letf∈C2q+p+2[-1,1], f(2q+p+2)∈BV[-1,1], q≥0, p≥1, and(59)f2k+1±1=0,k=0,…,q-1.Assume the polynomial(60)∑k=0ppkQrp-k2q+1+k!-1kxkhas only real-valued and nonzero roots x=zk, k=1,…,p, and let(61)θkc=θks=1-zkN,k=1,…,p.Then, the following estimate holds for x<1:(62)RN,pf,θc,θs,x=oN-2q-p-2,N⟶∞.Theorem7 is valid only if, for given p and q, 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 forΔNw(Δ^p(θ,f^c)) and ΔNw(Δ^p(θ,f^s)).Lemma 8.
Assumef∈C2q+p+(p+1)/2+2[-1,1], f(2q+p+(p+1)/2+2)∈BV[-1,1], q≥0, p≥1, and(63)f2k+1±1=0,k=0,…,q-1.Assume polynomial (60) has only real-valued and nonzero roots x=zk, k=1,…,p, and θk is defined by (26) with τk=zk. Let w≤p, when w and p have the same parity, and w≤p+1, otherwise. Then, the following estimates hold as N→∞:(64)ΔNwΔ^pθ,f^c=-1NNp∑j=0rcj∑t=2q+p+w-j+1/22q+p+1/2+11Nt+2∑s=qt-w/2A2s+1fπ2s+22s+1!σs,t,jw+oN-2q-p-p+1/2-3,(65)ΔNwΔ^pθ,f^s=-1N+1Np∑j=0rcj∑t=2q+p+w-j+1/22q+p+1/2+11Nt+2∑s=qt-w/2B2s+1fπ2s+22s+1!σ~s,t,jw+oN-2q-p-p+1/2-3,where(66)σs,t,jw=2q+p+1!∑k=0ppkβk,s,twp-k+t+1!2q+p+1-k!kj,σ~s,t,jw=2q+p+1!∑k=0ppkβ~k,s,twp-k+t+1!2q+p+1-k!kj,with β and β~ defined in Lemma 5.Proof.
We prove only (64). By taking n=N in (33) and using (56), we get(67)ΔNwΔ^pθ,f^c=-1NNp∑j=0rcj∑t=2q+w2q+p+1/2+11Nt+2∑s=qt-w/2A2s+1fπ2s+22s+1!σs,t,jw+oN-2q-p-p+1/2-3.This completes the proof in view of (B.15) and (B.16).Further, in Theorems9 and 10, we show that the pointwise convergence rate of the rational approximations depends on the asymptotic ΔN0(Δ^p(θ,f^c)) and ΔN0(Δ^p(θ,f^s)). From the other side, Lemma 8 reveals that the convergence rates of those sequences depend on the value of (as w=0)(68)p-j+12.When p is odd, for the highest power of 1/N, parameter j can be only j=0. It means that Qr(k)≡1. When p is even, parameter j can be j≤1 which means that Qr(k)=1+c1k. We determine parameter c1 later.Next theorem unveils the convergence rate of the MTR-approximations for odd values ofp, when Qr(k)=Q0(k)≡1. Note that, in this case, the roots of polynomial (60) coincide with the roots of generalized Laguerre polynomial Lp(2q+1)(x) (see Appendix A).Theorem 9.
Let parameterp≥1 be odd, f∈C2q+p+(p+1)/2+2[-1,1], q≥0, f(2q+p+(p+1)/2+2)∈BV[-1,1], and(69)f2k+1±1=0,k=0,…,q-1.Let θk, k=1,…,p, be defined by (26), where τk are the roots of the generalized Laguerre polynomial Lp(2q+1)(x). Then, the following estimates hold for x<1:(70)RN,pcf,θ,x=A2q+1f-1N+1N2q+p+p+1/2+2π2q+22p+1cos⁡πx/22N-p+1cosp+1πx/2σq,2q+p+1/2,00+cosπx/22N-p2cosp+2πx/2σq,2q+p+1/2,01+oN-2q-p-p+1/2-2,N⟶∞,RN,psf,θ,x=B2q+1f-1NN2q+p+p+1/2+2π2q+22p+1sin⁡πx/22N-pcosp+1⁡πx/2σ~q,2q+p+1/2,00+sinπx/22N-p-12cosp+2⁡πx/2σ~q,2q+p+1/2,01+oN-2q-p-p+1/2-2,N⟶∞,where σ and σ~ are defined in Lemma 8.Proof.
We use (18) to estimate RN,pc(f,θ,x). The error RN,ps(f,θ,x) can be estimated similarly.
Taking into account the fact thatθk→1 as N→∞, we estimate only the sums on the right-hand side of (18). An application of the Abel transformation to the sums of RN,pc(f,θ,x) leads to the following expansion of the error:(71)∑n=N+1∞Δnpθ,f^ce±iπnx=-e±iπN+1x1+e±iπxΔNpθ,f^c-e±iπN+1x1+e±iπx2ΔN1Δ^pθ,f^c-e±iπN+1x∑w=2p+1/2+1ΔNwΔ^pθ,f^c1+e±iπxw+1+11+e±iπxp+1/2+2∑n=N+1∞Δnp+1/2+2Δ^pθ,f^ce±iπnx.According to Lemma 5, we have(72)Δnp+1/2+2Δ^pθ,f^c=oN-pn2q+p+1/2+3,N⟶∞,n≥N+1,and hence the last term on the right-hand side of (71) is o(N-2q-p-(p+1)/2-2) as N→∞. It follows from Lemma 8 that the third term in (71) is O(N-2q-p-(p+1)/2-3) as N→∞. Therefore,(73)RN,pcf,θ,x=-eiπN+1x21+eiπxp+1+e-iπN+1x21+e-iπxp+1ΔNpθ,f^c-eiπN+1x21+eiπxp+2+e-iπN+1x21+e-iπxp+2ΔN1Δ^pθ,f^c+oN-2q-p-p/2-2.By Lemma 8, we have(74)ΔNpθ,f^c=-1NNp∑t=2q+p+1/22q+p+1/2+11Nt+2∑s=qt/2A2s+1fπ2s+2σs,t,00+oN-2q-p-p+1/2-3=-1NπN2q+2Np+p+1/2∑s=qq+p+1/4A2s+1fπ2sσs,2q+p+1/2,00+oN-2q-p-p+1/2-2,ΔN1Δ^pθ,f^c=-1NNp∑t=2q+p+1/22q+p+1/2+11Nt+2∑s=qt-1/2A2s+1fπ2s+2σs,t,01+oN-2q-p-p+1/2-3=-1NπN2q+2Np+p+1/2∑s=qq+p-1/4A2s+1fπ2sσs,2q+p+1/2,01+oN-2q-p-p+1/2-2.
Taking into account the fact that (see (B.16))(75)σs,2q+p+1/2,00=σs,2q+p+1/2,01=0,s>q,we conclude that σs,2q+(p+1)/2,0(0) and σs,2q+(p+1)/2,0(1) are nonzero only for s=q which leads to the following estimates:(76)ΔNpθ,f^c=A2q+1f-1NπN2q+2Np+p+1/2σq,2q+p+1/2,00+oN-2q-p-p+1/2-2,ΔN1Δ^pθ,f^c=A2q+1f-1NπN2q+2Np+p+1/2σq,2q+p+1/2,01+oN-2q-p-p+1/2-2.From here, we get(77)RN,pcf,θ,x=A2q+1f-1N+1π2q+2N2q+p+p/2+2σq,2q+p+1/2,00ReeiπN+1x1+eiπxp+1+σq,2q+p+1/2,01ReeiπN+1x1+eiπxp+2+oN-2q-p-p+1/2-2,which completes the proof.Figure3 shows the result of approximation of (53) by the MTR-approximations with optimal parameters τkc=τks, k=1,…,p as the roots of Lp(2q+1)(x). We see better accuracy on [-0.7,0.7] compared to nonoptimal parameters as in Figure 2. For p=1, the improvement is almost 25 times, and for p=3, the improvement is almost 240 times.Figure 3
The graphs of|RN,p(f,θc,θs,x)| on interval [-0.7,0.7] for N=64 while approximating (53) by MTR-approximations. Parameters τkc and τks, k=1,…,p, are the roots of Lp(2q+1)(x) which are optimal for odd p on |x|<1 (see Theorem 9).Next theorem deals with even values ofp. As we mentioned above, the best convergence rate is possible if Qr(k)=Q1(k)=1+c1k and τk, k=1,…,p, are the roots of (60). We need to assume that polynomial (60) has only real-valued and nonzero roots x=zk, k=1,…,p, for fixed p and q. In two cases, we can prove that it is true. When c1=0, the roots of polynomial (60) coincide with the roots of the generalized Laguerre polynomial Lp(2q+1)(x) (see Appendix A). When c1=-1/(2q+1+p), the roots coincide with the ones of Lp(2q)(x).We saw from our experiments (which we cannot prove theoretically) that polynomial (60) has only real-valued and nonzero roots also for other values of parameter c1. However, based on our experiments, we observed that the rational approximations have almost similar accuracy for different values of c1 while approximating smooth functions on x<1.Theorem 10.
Let parameterp≥2 be even, f∈C2q+p+p/2+2[-1,1], q≥0, f(2q+p+p/2+2)∈BV[-1,1], and(78)f2k+1±1=0,k=0,…,q-1.Assume the polynomial(79)∑k=0ppk1+c1p-k2q+1+k!-1kxkhas only real-valued and nonzero roots x=zk, k=1,…,p, and let θk be defined by (26) with τk=zk. Then, the following estimates hold for x<1:(80)RN,pcf,θ,x=A2q+1f-1N+1N2q+p+p/2+2π2q+22p+1cos⁡πx/22N-p+1cosp+1⁡πx/2σq,2q+p/2,00+σq,2q+p/2,10c1+cos⁡πx/22N-pσq,2q+p/2,112cosp+2⁡πx/2c1+oN-2q-p-p/2-2,N⟶∞,(81)RN,psf,θ,x=B2q+1f-1NN2q+p+p/2+2π2q+22p+1sin⁡πx/22N-pcosp+1⁡πx/2σ~q,2q+p/2,00+σ~q,2q+p/2,10c1+sin⁡πx/22N-p-1σ~q,2q+p/2,112cosp+2⁡πx/2c1+oN-2q-p-p/2-2,N⟶∞,where σ and σ~ are defined in Lemma 8.Proof.
We prove only (80) and need only to estimate the sums on the right-hand side of (18). Similar to (71), we apply the Abel transformation and get the following expansion of the error:(82)∑n=N+1∞Δnpθ,f^ce±iπnx=-e±iπN+1x1+e±iπxΔNpθ,f^c-e±iπN+1x1+e±iπx2ΔN1Δ^pθ,f^c-e±iπN+1x∑w=2p/2+1ΔNwΔ^pθ,f^c1+e±iπxw+1+11+e±iπxp/2+2∑n=N+1∞Δnp/2+2Δ^pθ,f^ce±iπnx.Taking into account Lemma 5, we obtain(83)Δnp/2+2Δ^pθ,f^c=oN-pn2q+p/2+3,N⟶∞,n≥N+1.And the last term on the right-hand side of (82) is o(N-2q-p-p/2-2) as N→∞. According to Lemma 8, the third term in the right-hand side of (82) is O(N-2q-p-p/2-3) as N→∞. Therefore,(84)RN,pcf,θ,x=-eiπN+1x21+eiπxp+1+e-iπN+1x21+e-iπxp+1ΔNpθ,f^c-eiπN+1x21+eiπxp+2+e-iπN+1x21+e-iπxp+2ΔN1Δ^pθ,f^c+oN-2q-p-p/2-2.According to Lemma 8, we get(85)ΔNpθ,f^c=-1NNp∑j=01cj∑t=2q+p/22q+p/2+11Nt+2∑s=qt/2A2s+1fπ2s+2σs,t,j0+oN-2q-p-p/2-3=-1NπN2q+2Np+p/2∑j=01cj∑s=qq+p/4A2s+1fπ2sσs,2q+p/2,j0+oN-2q-p-p/2-2,ΔN1Δ^pθ,f^c=-1NNp∑j=01cj∑t=2q+p+2-j/22q+p/2+11Nt+2∑s=qt-1/2A2s+1fπ2s+2σs,t,j1+oN-2q-p-p/2-3=-1Nc1πN2q+2Np+p/2∑s=qq+p-2/4A2s+1fπ2sσs,2q+p/2,11+oN-2q-p-p/2-2.
According to (B.16), we have(86)σs,2q+p+1/2,j0=σs,2q+p+1/2,j0=0,s>q,σs,2q+p+1/2,11=0,s>q.Hence, σs,2q+(p+1)/2,j(0),j=0,1 and σs,2q+(p+1)/2,1(1) are nonzero only for s=q, which leads to the following estimates:(87)ΔNpθ,f^c=A2q+1f-1NπN2q+2Np+p/2σq,2q+p/2,00+σq,2q+p/2,10c1+oN-2q-p-p/2-2,ΔN1Δ^pθ,f^c=A2q+1f-1Nc1πN2q+2Np+p/2σq,2q+p/2,11+oN-2q-p-p/2-2.
Finally, from (84), we get(88)RN,pcf,θ,x=A2q+1f-1N+1π2q+2N2q+p+p/2+2σq,2q+p/2,00+σq,2q+p/2,10c1ReeiπN+1x1+eiπxp+1+σq,2q+p/2,11ReeiπN+1x1+eiπxp+2c1+oN-2q-p-p/2-2,which completes the proof.Theorems9 and 10 conclude that by appropriate determination of parameters τkc and τks, k=1,…,p, we get extra improvement of the convergence rate of the MTR-approximations by factor O(N(p+1)/2) compared to Theorem 6. Final improvement compared to the modified expansion is by factor O(Np+(p+1)/2).Let us return to MTR-approximations of (53). Figure 4 shows the results of approximation of (53) by the MTR-approximations with even p. Parameters τkc=τks, k=1,…,p are selected as the roots of Lp(2q+1)(x). Compared with Figure 2, we see better accuracy on x<1. For p=2, the improvement is almost 27 times, and for p=4, the improvement is almost 200 times.Figure 4
The graphs ofRN,pf,θc,θs,x on interval [-0.7,0.7] for N=64 while approximating (53). Parameters τkc and τks, k=1,…,p, are the roots of Lp(2q+1)(x).Figure5 shows similar results with parameters τkc=τks, k=1,…,p, as the roots of Lp(2q)(x). In the next section, we will prove that those parameters provide improved accuracy also at x=±1 for some p and q. We see that both choices of parameters provide similar results on x<1.Figure 5
The graphs ofRN,pf,θc,θs,x on interval [-0.7,0.7] for N=64 while approximating (53). Parameters τkc and τks, k=1,…,p, are the roots of Lp(2q)(x).


## 4. Pointwise Convergence at the Endpoints

In this section, we explore the pointwise convergence of the MTR-approximations at the endpointsx=±1. Next theorem explores the convergence of the rational approximations without determining parameters τc and τs.Theorem 11.
Assumef∈C2q+p+1[-1,1], f(2q+p+1)∈BV[-1,1], q≥0, p≥1, and(89)f2k+1±1=0,k=0,…,q-1.Let θk, k=1,…,p be defined by (26). Then, the following estimates hold:(90)RN,pcf,θ,±1=A2q+1fN2q+1-1p2q+1!π2q+2γpτhp,2qτ+oN-2q-1,N⟶∞,RN,psf,θ,±1=±B2q+1fN2q+1-1p2q+1!π2q+2γpτhp,2qτ+oN-2q-1,N⟶∞,where hp,m(τ) is defined by (46).Proof.
In view of (18) and (19), we write(91)RN,pcf,θ,±1=Npγpτ∑n=N+1∞Δnpθ,f^c-1n,(92)RN,psf,θ,±1=∓Npγpτ∑n=N+1∞Δnpθ,f^s-1n.
Now, the proof immediately follows from the estimates of Lemma5 by taking w=0,r=0 and by recalling that αp-k,p-k=(-1)p-k(p-k)!.Note that this theorem is valid also forp=0 which corresponds to the expansions by the modified Fourier basis (compare with Theorems 1 and 2).Exact constants of the main terms in (90), for p=0, can be found also in [10] (Theorem 3.2). We see that, in general, rational corrections do not increase the convergence rates of modified Fourier expansions at the endpoints x=±1 without specifying appropriately parameters τc and τs. Both approaches have the same convergence rates O(N-2q-1).Moreover, as Figure6 shows, without reasonable selection of the parameters, modified Fourier expansions have better accuracy compared to “nonoptimal” rational approximations at x=±1.Figure 6
The graphs ofRN,pf,θc,θs,x for p=0,1,2,3 and N=64 while approximating (53) at the points x=±1. In rational approximations, we took τkc=τks=k, k=1,…,p.Is it possible to improve the accuracy by appropriate selection of parametersτc and τs? The answer is positive and the solution is in the estimates of Theorem 11. Like the previous section, we put(93)γkτ=pk2q+p!2q+p-k!Prk,where(94)Prk=∑j=0rdjkj,d0=1.Now, the property hp,2q(τ)=0 follows from the identity(95)∑k=0p-1kpkkj=0,j<p.Next theorem is the result of these observations and Theorem11.Theorem 12.
Letf∈C2q+p+1[-1,1], q≥0, p≥1, f(2q+p+1)∈BV[-1,1], and(96)f2k+1±1=0,k=0,…,q-1.Assume the polynomial(97)∑k=0ppkPrp-k2q+k!-1kxkhas only real-valued and nonzero roots x=zk, k=1,…,p, and let(98)θkc=θks=1-zkN,k=1,…,p.Then,(99)RN,pf,θc,θs,±1=oN-2q-1,N⟶∞.Note that Theorem12 is valid only for those parameters p and q when polynomial (97) has only real-valued and nonzero roots. Further, we clarify this property with more details.Our next goal is derivation of the exact convergence rate of (99).Lemma 13.
Assumef∈C2q+p+[(p+1)/2]+1[-1,1], q≥0, p≥1, f(2q+p+[(p+1)/2]+1)∈BV[-1,1], and(100)f2k+1±1=0,k=0,…,q-1.Assume the polynomial(101)∑k=0ppkPrp-k2q+k!-1kxkhas only real-valued and nonzero roots x=zk, k=1,…,p, and let θk be defined by (26) with τk=zk. Then, the following asymptotic expansions hold:(102)RN,pcf,θ,±1=1γpτ∑j=0rdj∑t=2q+p+1-j/22q+p+1/21Nt+1∑l=0t-2qbl-1ll!∑s=qt-l/2A2s+1fπ2s+22s+1!δq,t-l,jl-∑s=qt/2A2s+1fπ2s+22s+1!δs,t,j1+oN-2q-p+1/2-1,N⟶∞,(103)RN,psf,θ,±1=±1γpτ∑j=0rdj∑t=2q+p+1-j/22q+p+1/21Nt+1∑l=0t-2qbl-1ll!∑s=qt-l/2B2s+1fπ2s+22s+1!δ~q,t-l,jl-∑s=qt/2B2s+1fπ2s+22s+1!δ~s,t,j1+oN-2q-p+1/2-1,N⟶∞,where bl is the lth Bernoulli number and(104)δs,t,jw=2q+p!∑k=0ppkβk,s,t0p-k+t+w!2q+p-k!kj,(105)δ~s,t,jw=2q+p!∑k=0ppkβ~k,s,t0p-k+t+w!2q+p-k!kj,with β and β~ defined in Lemma 5.Proof.
We prove only (102). In view of Lemma 5 (with w=0) and (91), we write(106)RN,pcf,θ,±1=Npγpτ∑k=0pγkτNk∑t=2q2q+p+1/2p-k+t+1!∑s=qt/2A2s+1fπ2s+22s+1!βk,s,t0∑n=N+1∞1np-k+t+2+oN-2q-p+1/2-1. We estimate the infinite sum on the right-hand side of (106) by the Euler-Maclaurin formula (see [21]). We have(107)∑n=N∞1np-k+t+2=1p-k+t+1∑w=02q+p+1/2-tp-k+t+wwbw-1wNp-k+t+w+1+ON-2q-p+k-p+1/2-2,where bw is the wth Bernoulli number. Then,(108)RN,pcf,θ,±1=1γpτ∑j=0rdj∑t=2q2q+p+1/21Nt+1∑l=0t-2qbl-1ll!∑s=qt-l/2A2s+1fπ2s+22s+1!δs,t-l,jl-1γpτ∑j=0rdj∑t=2q2q+p+1/2-11Nt+2∑r=qt/2A2s+1fπ2s+22s+1!δs,t,j1+oN-2q-p+1/2-1,which completes the proof in view of (B.2) and (B.3) (see Appendix B).By repeating the observations of previous section, it is possible to deduce that, for getting the maximal convergence rate for odd values ofp, the polynomial Qr(k) can be at most degree 0 polynomial, Qr(k)=Q0(k)≡1. For even values of p, Qr(k)=Q1(k)=1+d1k. In the first case, parameters τk are the roots of Lp(2q)(x). In the second case, if d1=0, we get the roots of Lp(2q)(x) and if d1=-1/(2q+p), we get the roots of Lp(2q-1)(x). The next two theorems immediately follow from Lemma 13 and identity (B.12) and, we omit the proofs.Theorem 14.
Let parameterp≥1 be odd, f∈C2q+p+(p+1)/2+2[-1,1], q≥0, f(2q+p+(p+1)/2+2)∈BV[-1,1], and(109)f2k+1±1=0,k=0,…,q-1.Let θk, k=1,…,p, be defined by (26), where τk, k=1,…,p, are the roots of the generalized Laguerre polynomial Lp(2q)(x). Then, the following estimates hold as N→∞:(110)RN,pcf,θ,±1=1N2q+p+1/2+1A2q+1fγpτπ2q+22q+1!δq,2q+p+1/2,00-12δq,2q+p-1/2,01+oN-2q-p+1/2-1,RN,psf,θ,±1=±1N2q+p+1/2+1B2q+1fγpτπ2q+22q+1!δ~q,2q+p+1/2,00-12δ~q,2q+p-1/2,01+oN-2q-p+1/2-1,where δ and δ~ are defined in Lemma 13.Figure7 shows the result of approximation of (53) by the rational approximations with optimal values of parameters τk, k=1,…,p, as in Theorem 14. We see that, by increasing p (p is odd), we increase the accuracy of approximations at the points x=±1. Note that p=0 corresponds to the classical expansion by the modified basis and we see that, in contrary to Figure 6, the optimal choice of parameters do have big positive impact on the accuracy.Figure 7
The values of-log10⁡max⁡RN,pf,θc,θs,±1 while approximating (53) for different N and p. Parameters τkc and τks, k=1,…,p, are the roots of the generalized Laguerre polynomial Lp(2q)(x).Comparison of Theorems9 and 14 reveals the problem of the optimal rational approximations which is in the difference of optimal values of parameters τk for x<1 and x=±1. On x<1 and x=±1, the optimal values are the roots of Lp(2q+1)(x) and Lp(2q)(x), respectively. The choice of Lp(2q)(x) will result in better accuracy on overall [-1,1] by the uniform norm, but on x<1 the rate of convergence will be worse by factor O(N).Next theorem explores even values ofp.Theorem 15.
Let parameterp≥2 be even, f∈C2q+p+p/2+2[-1,1], q≥0, f(2q+p+p/2+2)∈BV[-1,1], and(111)f2k+1±1=0,k=0,…,q-1.Assume the polynomial(112)∑k=0ppk1+d1p-k2q+k!-1kxkhas only real-valued and nonzero roots x=zk, k=1,…,p, and let θk be defined by (26) with τk=zk. Then, the following asymptotic expansions hold:(113)RN,pcf,θ,±1=1N2q+p/2+1A2q+1fγpτπ2q+22q+1!Φq,pd1+oN-2q-p/2-1,N⟶∞,(114)RN,psf,θ,±1=±1N2q+p+1/2+1B2q+1fγpτπ2q+22q+1!Φ~q,pd1+oN-2q-p/2-1,N⟶∞,where(115)Φq,pd1=δq,2q+p/2,00-δq,2q+p/2-1,01+d1δq,2q+p/2,10-12δq,2q+p/2-1,11,Φ~q,pd1=δ~q,2q+p/2,00-δ~q,2q+p/2-1,01+d1δ~q,2q+p/2,10-12δ~q,2q+p/2-1,11,with δ and δ~ defined in Lemma 13.Estimates (113) and (114) are valid if polynomial (112) has only real-valued and nonzero roots. As we mentioned above, in two particular cases when d1=0 and d1=-1/(2q+p), the roots of the polynomial coincide with the roots of Lp(2q)(x) and Lp(2q-1)(x), respectively. Both Laguerre polynomials have only real-valued and positive roots and Theorem 15 is valid in both cases. The choice of polynomial Lp(2q)(x) is reasonable as it will provide optimal rational approximation both on x<1 (see Theorem 10) and at x=±1 for some p and q.Figure8 shows the result of application of the rational approximations to function (53) with optimal values of parameters τk, k=1,…,p, as the roots of Lp(2q)(x).Figure 8
The values of-log10⁡max⁡RN,pf,θc,θs,±1 while approximating (53) for different N and p. Parameters τkc and τks, k=1,…,p, are the roots of the generalized Laguerre polynomial Lp(2q)(x).Now, we show that, for somep and q, estimates (113) and (114) can be improved with appropriate selection of parameter d1. Assume that, for given p and q, it is possible to vanish Φq,p(d1) by appropriate selection of d1 in (113). Then, we derive(116)RN,pcf,θc,θs,±1=oN-2q-p/2-1,or(117)RN,pcf,θc,θs,±1=ON-2q-p/2-2in case of smoother functions. Similarly, if Φ~q,p(d1)=0 by appropriate selection of parameter d1, then,(118)RN,psf,θc,θs,±1=oN-2q-p/2-1,or(119)RN,psf,θc,θs,±1=ON-2q-p/2-2in case of smoother functions.The problem is that we cannot vanish bothΦq,p(d1) and Φ~q,p(d1) simultaneously by the same d1. Hence, we decompose a function into even and odd parts and perform separate optimizations in terms of parameter d1. In order to choose parameter d1 appropriately, we need to have(120)δq,2q+p/2,10-12δq,2q+p/2-1,11≠0,(121)δ~q,2q+p/2,10-12δ~q,2q+p/2-1,11≠0for even and odd functions, respectively.Then, we put(122)d1=dpevenq=-δq,2q+p/2,00+δq,2q+p/2-1,01δq,2q+p/2,10-1/2δq,2q+p/2-1,11,d1=dpoddq=-δ~q,2q+p/2,00+δ~q,2q+p/2-1,01δ~q,2q+p/2,10-1/2δ~q,2q+p/2-1,11.Finally, we put dpeven(q) and dpodd(q) into (112) and if that polynomials have only real-valued and nonzero roots, the optimization process will succeed. Tables 1 and 2 show that, except some special cases, we can optimize estimates of Theorem 15.Table 1
The values ofdpeven(q) and the roots of (112) for p=2,4,6 and 0≤q≤6.
q 0 1 2 3 4 5 6 d2even(q) - 1 1 1 / 3 1 / 5 1 / 7 1 / 9 1 / 11 τ1 - 1.41 2.71 4.26 5.89 7.57 9.28 11.01 τ2 1.41 13.29 11.74 13.31 15.29 17.39 19.53  d4even(q) - 3 / 14 - 3 / 2 3 / 10 3 / 22 3 / 34 3 / 46 3 / 58 τ1 0.12 - 27.44 2.61 3.81 5.10 6.45 7.85 τ2 1.09 1.57 5.86 7.50 9.23 10.99 12.78 τ3 3.46 4.44 10.85 12.72 14.76 16.86 18.98 τ4 7.91 9.43 22.28 21.43 23.15 25.36 27.71  d6even(q) - 1 / 11 - 1 / 5 1 1 / 7 1 / 13 1 / 19 1 / 25 τ1 0.18 - 0.74 1.96 2.90 3.95 5.07 6.25 τ2 1.00 1.34 4.28 5.57 6.98 8.46 9.98 τ3 2.60 3.41 7.51 9.04 10.76 12.54 14.35 τ4 5.15 6.45 11.99 13.62 15.55 17.59 19.67 τ5 8.96 10.74 18.54 19.91 21.87 24.08 26.38 τ6 14.84 17.18 75.72 31.25 31.35 33.32 35.69Table 2
The values ofdpodd(q) and the roots of (112) for p=2,4,6 and 0≤q≤4.
q 0 1 2 3 4 5 6 d2odd(q) - - 1 / 2 1/4 1 / 6 1 / 8 1 / 10 τ1 - - 4.417 6 7.653 9.347 11.07 τ2 - - 13.58 14 15.68 17.65 19.73  d4odd(q) - 3 / 20 - 3 / 8 3 / 4 3 / 16 3 / 28 3 / 40 3 / 52 τ1 0.23 - 1.95 2.70 3.87 5.14 6.48 7.87 τ2 1.35 1.83 6.09 7.63 9.31 11.05 12.83 τ3 3.77 4.96 11.41 13.01 14.93 16.98 19.07 τ4 8.249 10.16 35.80 22.99 23.76 25.69 27.92  d6odd(q) - 1 / 14 - 1 / 8 - 1 / 2 1 / 4 1 / 10 1 / 16 1 / 22 τ1 0.20 0.53 - 16.59 2.94 3.98 5.09 6.27 τ2 1.06 1.80 2.07 5.67 7.04 8.50 10.01 τ3 2.71 3.88 4.52 9.23 10.86 12.60 14.40 τ4 5.29 6.92 7.93 13.95 15.73 17.70 19.74 τ5 9.14 11.23 12.62 20.58 22.21 24.28 26.52 τ6 15.04 17.65 19.45 37.63 32.58 33.82 35.98Figures9 and 10 show the errors at x=±1 while approximating (53) with rational approximations, where parameters τkc and τks, k=1,…,p, are selected according to Tables 1 and 2 for even and odd parts of the function, respectively. We called this approach “optimal” in the figures. For comparison, we showed also the result of approximations with parameters τkc and τks, k=1,…,p, as the roots of Lp(2q)(x) (see Figure 8 and Theorem 15). We called the latest “nonoptimal” in the figures. We see the impact of optimizations on the accuracy of the rational approximations at x=±1.Figure 9
The values of-log10⁡max⁡RN,pf,θc,θs,±1 for p=4 and different N while approximating (53). In “nonoptimal” case, parameters τkc and τks, k=1,…,p, are the roots of Lp(2q)(x). In “optimal” case, the parameters are chosen from Tables 1 and 2 for p=4 and q=1.Figure 10
The values of-log10⁡max⁡RN,pf,θc,θs,±1 for p=6 and different N while approximating (53). In “nonoptimal” case, parameters τkc and τks, k=1,…,p, are the roots of Lp(2q)(x). In “optimal” case, the parameters are chosen from Tables 1 and 2 for p=6 and q=1.Throughout the paper, we systematically required that approximated functionf obeys first derivative conditions (8). Without those conditions, the convergence rate will remain slow. This is due to function jumps in certain derivatives at the endpoints x=±1. If these jumps are known, the convergence acceleration can be achieved by well-known polynomial subtraction approach. For the classical Fourier series this approach has a very long history (see [3, 22–28]). For modified expansions this approach is explored in [3, 5, 7]. More specifically, we write f (see [3]) in the terms of its Lanczos representation:(123)f=f-gk+gk,where functions (polynomials) gk are chosen as such to satisfy the conditions:(124)f2r+1±1=gk2r+1±1,r=0,…,k-1.Since f-gk obeys the first k derivative conditions, the new approximation(125)MNkf,x=MNf-gk,x+gk will converge with the same rate as if f obeyed those conditions. This is the polynomial subtraction technique known also as Krylov-Lanczos approach. If the jumps of f are unknown, their values can be approximated by solution of the corresponding system of linear equations (see [22]).The same approach can be applied also for the MTR-approximations and in that case all theorems of this paper will remain valid without the requirements of the derivatives at the endpointsx=±1.


## 5. Conclusion

In this paper, we investigated the convergence of the MTR-approximationsMN,p with parameters θc and θs defined by (24). The main goal of the paper was to show that by appropriate selection of parameters τc and τs it was possible to improve substantially the pointwise convergence of the approximations. We accomplished the main goal by calculating the exact constants of the main terms of asymptotic errors and by eliminating those constants through appropriate selection of the approximation parameters. We showed that the optimization depends whether x<1 or x=±1 and also whether parameter p is odd or even.Theorem6 provides general estimate on x<1 proving that, without optimal selection of parameters τkc and τks, k=1,…,p, the rational approximations have convergence rate O(N-2q-p-2) as N→∞ if an approximated function has enough smoothness and obeys the first q derivative conditions (see (8)). Compared with the modified Fourier expansions (see Theorems 1 and 2 or the same theorem with p=0), the improvement is by factor O(Np) as N→∞.Theorem9 gave the optimal choice for parameters τk when x<1 and p was odd. If τkc=τks, k=1,…,p, were the roots of the generalized Laguerre polynomial Lp(2q+1)(x) then the rational approximations had convergence rate O(N-2q-p-(p+1)/2-2) with improvement by factor O(N(p+1)/2) compared to nonoptimal choice of parameters (Theorem 6). The improvement was by factor O(N(p+1)/2+p) compared to the expansions by the modified Fourier basis.In case of even values ofp and x<1 (Theorem 10), possible selection set of optimal parameters is wider. If for given p and q the polynomial(126)∑k=0ppk1+c1p-k2q+1+k!-1kxkhas only nonzero and real-valued roots x=zk, k=1,…,p, then, selection τks=τkc=zk provides better convergence rate O(N-2q-p-p/2-2) compared to the estimate of Theorem 6 and improvement is by factor O(Np/2). Improvement is by factor O(Np/2+p) compared to the expansions by the modified Fourier basis. The problem is to find the values of c1 in (126) for which it will have only real-valued and nonzero roots. In two cases it is obvious. When c1=0, the roots of (126) coincide with the roots of Lp(2q+1)(x). When c1=-1/(2q+p+1), the roots coincide with the ones of Lp(2q)(x). In both cases all roots are positive.Theorem11 imparts the convergence rate O(N-2q-1) of the rational approximations at x=±1 without optimal selection of parameters. Comparison with Theorems 1 and 2 shows no improvement. Moreover, as our experiments show (see Figure 6) rational approximations without reasonable selection of parameters τk can perform worse at x=±1 compared to the expansions by the modified Fourier basis.Theorem14 found the optimal values of parameters for odd p for better convergence rate at x=±1. It proved that the best accuracy could be achieved when parameters τks=τkc were the roots of the generalized Laguerre polynomial Lp(2q)(x). For that choice, the convergence rate was O(N-2q-(p+1)/2-1) and improvement was by factor O(N(p+1)/2) compared to the modified Fourier expansions.We see that whenp is odd, the optimal choices for x<1 and x=±1 are different. The choice of polynomial Lp(2q)(x) will provide the minimal uniform error on [-1,1], but, for x<1, the convergence rate will be worse by factor O(N) compared to the optimal choice Lp(2q+1)(x).In case of evenp, we obtained similar results. Theorem 15 outlines the set of optimal parameters. If for given p and q the polynomial(127)∑k=0ppk1+d1p-k2q+k!-1kxkhas only real-valued and nonzero roots x=zk for some d1, then, selection τkc=τks=zk will provide convergence rate O(N-2q-p/2-1) with improvement by factor O(Np/2) compared to the modified Fourier expansions. The problem is the same as that for polynomial (126). Polynomial (127) must have only real-valued and nonzero roots for the selected d1. Fortunately, two such selections are known. When d1=0 or d1=-1/(2q+p), the roots of (127) coincide with the ones of Lp(2q)(x) and Lp(2q-1)(x), respectively. The choice of Lp(2q)(x) is better as it will provide optimal approximations for both x<1 and x=±1. However, estimates of Theorem 15 allow determining parameter d1 for even more better convergence rate. Tables 1 and 2 show some values of d1 and parameters τk that will provide convergence rate O(N-2q-p/2-2) with improvement by factor O(N). The problem is that the latest choice is not optimal for x<1. It will give worse accuracy compared to the optimal selection for x<1. Convergence rate will degrade by factor O(N). As in case of odd p, a user of the algorithms must decide which choice will be more appropriate to select, the best approximation on overall [-1,1] with worse accuracy on x<1, or the best accuracy on the latest one.



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