On the Convergence of the Quasi-Periodic Approximations on a Finite Interval

. We investigate the convergence of the quasi-periodic approximations in diﬀerent frameworks and reveal exact asymptotic estimates of the corresponding errors. The estimates facilitate a fair comparison of the quasi-periodic approximations to other classical well-known approaches. We consider a special realization of the approximations by the inverse of the Vander-monde matrix, which makes it possible to prove the existence of the corresponding implementations, derive explicit formulas and explore convergence properties. We also show the application of polynomial corrections for the convergence acceleration of the quasi-periodic approximations. Numerical experiments reveal the auto-correction phenomenon related to the polynomial corrections so that utilization of approximate derivatives surprisingly results in better convergence compared to the expansions with the exact ones.


Introduction
We continue investigations of the reconstruction problem of a smooth but non-periodic function f on [−1, 1] by the finite number of its Fourier coefficients It is well known that the solution of the problem by the truncated Fourier series or trigonometric interpolation via corresponding Fourier coefficients is ineffective for non-periodic functions due to slow pointwise convergence on (−1, 1) and the Gibbs phenomenon at the endpoints (see [1][2][3][4][5]). The degraded convergence may imply an impression that the accurate reconstruction of f from the knowledge of the coefficients is impossible. However, many authors proved the opposite by showing that the spectral information is sufficient even for non-periodic functions. There is ample literature devoted to the resolution of the Gibbs phenomenon and convergence accelerations. It is impossible to summarize such a great amount of work. Individual articles include . We will follow only the ideas related to the current paper. An efficient approach to convergence acceleration by subtracting a polynomial representing the discontinuities (jumps) in the function and its derivatives was suggested by Krylov [57] (see also [58]). Lanczos [59] independently developed the same approach by introducing the basic system of 2-periodic Bernoulli polynomials. We will refer to this approach as the Krylov-Lanczos method. Jones and Hardy [60] and Lyness [61] considered convergence acceleration of trigonometric interpolations by the polynomial subtraction (see also [62,63]).
The key problem in the Krylov-Lanczos method is approximation of the jumps. In the case of interpolations, as the pointwise values of the function are known, the finite difference formulas can be applied for approximation of the derivatives in jumps. Approaches resembling this approach have been attempted in a series of papers [59,61,[64][65][66]. However, the approximation of jumps via finite differences is not recommended due to numerical instability (see for example [61]). Even in the case of a uniform grid, finite difference approximations are notoriously unreliable. Moreover, in many applications, the pointwise values are not explicitly available.
The first attempt towards a more robust approach was initiated in [67][68][69][70][71][72][73]. The general approach was established by Eckhoff in a series of papers [74][75][76], showing that coefficients contain sufficient information to reconstruct the jump-values. We will refer to this approach as the Eckhoff method, where the fundamental aspect is the approximation of the jumps by the solution of a linear system of equations. A similar idea was used in the Richardson extrapolation process [77]. Investigations and generalizations of the Krylov-Lanczos and the Eckhoff methods see also in [78][79][80][81][82][83][84][85][86][87][88][89]. Polynomial corrections applied to trigonometric expansions for the approximation/interpolation of the piecewise-smooth functions completely eliminate the Gibbs phenomenon and accelerate convergence both away from the singularities and on the entire interval. The main difficulty of its realization is the detection of singularities and the approximation of the corresponding jumps.
Another approach that does not eliminate the Gibbs phenomenon but mitigates its impacts utilizes expansions with trigonometric functions whose periods are larger than the lengths of approximation or interpolation in-tervals. The idea of such expansions was introduced in [90]. It considered trigonometric interpolation of a function f on [−1, 1] for the following basisfunctions {e iπnσx }, |n| ≤ N, σ = 2N 2N + m + 1 , m = 0, 1, 2, . . . , and for the grid x k ∈ [−1, 1], that includes the endpoints of the interval. All basis functions have the period 2/σ, which tends to 2 (the length of the interval of interpolation) as N tends to infinity. That is why those expansions were called as quasi-periodic (QP-) interpolations. Let us introduce the idea of [90] in more details (see also [91]). Consider a new function g(t) defined on [−σ, σ] by the following change of variable This implies interpolation of g(t) on the grid t k = σx k = 2k 2N + m + 1 , |k| ≤ N.
Thus, the QP-interpolation actually interpolates g(t) on the grid t k and is exact for e iπnt , |n| ≤ N . It is important to note that for m > 0, the 2-periodic extension of the grid t k to the real line is non-uniform. This non-uniformity should be one of the main reasons of better convergence properties of the QP-interpolation compared to the classical trigonometric interpolation that uses uniform grids. Papers [91][92][93][94][95] investigated the pointwise convergence of the QP-interpolation and its convergence in the L 2 -norm. The application of polynomial corrections to the QP-interpolation was considered in [95] for complete elimination of the Gibbs phenomenon and additional convergence acceleration. Similar ideas were applied to the truncated Fourier series in [96]. The corresponding expansions were named as QP-approximations. The main idea was the extension of a function f from the interval [−1, 1] onto the larger interval [−1/σ, 1/σ] and its approximation via truncated Fourier series with quasi-periodic basis-functions. The main problem in the realization of the idea is the determination of the coefficients on the extended interval, where the function f was unknown. Paper [96] described three algorithms A, B, and C that differently solved the problem. Moreover, it showed how the required coefficients could be approximated via classical Fourier coefficients known for the interval [−1, 1].
The main goal of the current paper is the investigation of the convergence of Algorithm C. The main benefit of the algorithm is its explicit realization that allows us to perform theoretical investigations regarding the existence of the solution and convergence properties. Further, we show how the polynomial correction method can be applied for additional convergence acceleration.
Approximations with non-periodic basis functions were also considered in [97][98][99]. However, they were essentially different by using the classical Fourier coefficients on [−1, 1] rather than the Fourier quasi-periodic coefficients on the extended interval.
Another approach that utilized non-periodic basis functions was introduced in [100,101]. Here, the truncated Fourier series by the following modified trigonometric system was considered It was originally proposed by [102] without investigation of its properties. System H is an orthonormal basis of L 2 [−1, 1] as it consists of the eigenfunctions of the Sturm-Liouville operator with Neumann boundary conditions f (1) = f (−1) = 0. The corresponding univariate and multivariate approximations were thoroughly investigated in a series of papers [103][104][105][106]. Some of the basis functions sinπ(n− 1 2 )x are nonperiodic on [−1, 1]. The rest of the basis functions cos πnx are 2-periodic. The non-periodicity implies some slight improvements in convergence compared to the classical Fourier series on [−1, 1]. However, the convergence rate of the modified Fourier approximation remains relatively slow. If N is the truncation parameter, the uniform error is O(N −1 ) on [−1, 1] and O(N −2 ) away from the endpoints. Better convergence results were achieved only via application of polynomial corrections [107,108] or rational trigonometric functions [109][110][111]. Interpolations by the modified trigonometric system were considered in [112]. Similarly, the convergence rate compared to the classical trigonometric interpolation was improved. Overall, without additional convergence acceleration, the expansions via modified trigonometric system have worse convergence compared to the QP-approximations and interpolations. One of our future works will be application of the ideas of the quasi-periodic expansions to the modified trigonometric system.
The idea of non-periodic approximations or interpolations is not new. Many authors considered similar ideas of approximation on extended domains [−T, T ], T > 1. Thus, they sought an approximation F N (f ) to f from the set Papers [113][114][115] proposed to compute F N (f ) as the best approximation to f on [−1, 1] in a least squares sense (named as continuous extension): where · is the standard norm on L 2 (−1, 1). In [115][116][117] it was shown that this approach leads to geometrically fast approximations for analytic functions. Regarding parameter T , those papers recommended T = 2 as the best choice, and also for oscillatory functions, where e tol << 1. The latest resembles some similarity to our approach as parameter 1/σ = T also tends to 1 as N → ∞. However, the constructions are essentially different from the QP-approximations by solving different linear systems of equations and utilizing different frameworks for optimality. One of our future works will be devoted to the convergence analysis of analytical functions and comparison with those known results. Another important topic should be stability analysis of the QPapproximations and comparison with the stabilities of known methods. More general problem was considered in [118,119] dealing with the reconstruction of piecewise-smooth functions on a finite interval via Fourier data. This is the most general approach, where the points of discontinuities with the corresponding jumps are unknown. Similar problem with combination of the QP-approximations will be considered elsewhere. In the current paper, we explore the simplest case with a discontinuity point at the endpoints of the interval (the case of non-periodic function), where the only problem is the determination of the corresponding jumps according to the ideas of Eckhoff.
The paper is organized as follows. Section 1 introduces QP-approximations and specifically Algorithm C. Section 2 proves some preliminary results. Section 3 explores the pointwise convergence away from the endpoints and at x = ±1. Section 4 deals with L 2 -convergence on the entire interval. In both cases, we derive exact asymptotic estimates for the corresponding errors. Section 5 considers convergence acceleration of the QP-approximations via polynomial corrections. It also studies the methods of approximation of function derivatives at the endpoints of the interval. It shows that application of approximate derivatives improves the convergence properties compared to the expansions that use the exact values of the derivatives. This improvement is known as auto-correction phenomenon which was previously detected and explained for polynomial corrections applied to trigonometric approximations and interpolations [86,87].

Quasiperiodic Approximations
Let a function f be defined on [−1, 1]. The main idea of the QP-approximations [96] is the extension of f from [−1, 1] onto the larger interval [−1/σ, 1/σ] and the application of the truncated Fourier series with the quasi-periodic basis functions and f * stands for the extension of f . Note that m = −1 corresponds to the truncated Fourier series of f on the interval [−1, 1]. It is essential to observe that σ in Equation (6)  . We put and split the coefficients (6) into three terms where We see that the main problem of the realization of the QP-approximations is the calculation of the first and the third integrals in the right-hand side of (7) as functions f lef t and f right are unknown. Paper [96] considered three different approaches (Algorithms A, B and C) for the extensions. Algorithm A defines f lef t and f right explicitly as a linear combination of some "supporting" functions with unknown coefficients and considers a system of linear equations for their determination. Algorithms B and C perform implicit extension and approximate the mentioned integrals in (7) via some quadrature formulae. The unknown values of the left and right extensions on the corresponding grids of the quadratures can be found as a solution to a system of linear equations. Algorithm B applies Gaussian quadrature and Algorithm C uses the simplest rectangular quadrature.
Let us explain Algorithm C in more details. Let be the grid of the rectangular quadrature on [1, 1/σ], and be the corresponding weights. The application of the rectangular rule to the first and the third integrals in the right-hand side of (7) implies Denoting we rewrite (8) as follows Paper [96] proposed determination of unknowns {c lef t j } m j=0 and {c right j } m j=0 from the following system of linear equations We hoped that (10) would assure a smooth and periodic continuation outside of [−1, 1] resulting in accelerated convergence of the corresponding truncated Fourier series on the extended interval. Throughout the paper, we only consider Algorithm C as it allows us to solve explicitly the system (10) through the inverse of a Vandermonde matrix. Let us derive it (see also [96]). For = 0, . . . , m, we have After changing the summation orders in the first sums, we get . . , 2m + 1, we rewrite the previous system as follows with unknowns {c k } 2m+1 k=0 and with the following Vandermonde matrix The solution of (11) can be derived explicitly through the inverse of the Vandermonde matrix (12). Let The standard technique for the calculation of the inverse (see details in [81] for a similar problem) implies where γ j are the coefficients of the following polynomial Now, we get the following explicit form of the coefficients F n,m in the terms of the inverse Algorithm C has several benefits. First and foremost, it provides the existence and uniqueness of the solution to the corresponding system of linear equations (see (10)). Second, practical solution can be achieved also by using the well-known Björk-Pereyra algorithm [120] for the solution to systems with Vandermonde matrices. This O((2m + 2) 2 ) algorithm has a number of beneficial properties. In particular, under certain mild hypotheses [121], the magnitude of the numerical errors depends only on the machine precision used, and is independent of condition numbers of matrices. However, for the theoretical purposes, we will use the explicit solution.

Preliminaries
We consider some preliminary statements and lemmas for the main theorems.
Consider a sequence of complex numbers {y s } ∞ s=−∞ and denotê It is easy to verify that δ p n (ŷ) = ∆ 2p n+p (ŷ), and Consequently, Then, for any n, the following estimate holds as N → ∞ Then, for any N , the following estimate holds as n → ∞ Proof. The proof is straightforward by means of integration by parts.
The next lemma uses two-point Taylor expansions.
Theorem 1 [122] Let f (z) be an analytic function on an open set Ω ⊂ C and z 1 , z 2 ∈ Ω with z 1 = z 2 . Then, f (z) admits the two-point Taylor expansion where the coefficients a n (z 1 , z 2 ) and a n (z 2 , z 1 ) are given by Cauchy integral The remainder term r N (z 1 , z 2 ; z) is given by the Cauchy integral The contour of integration C is a simple closed loop which encircles the points z 1 and z 2 (for a n ) and z, z 1 and z 2 (for r N ) in the counterclockwise direction and is contained in where Representation (18) is inappropriate for numerical computations. A more practical formula to compute the coefficients of the above two-point Taylor expansion is given in the following proposition.
Then, the remainder r p (Φ j , z) of the corresponding Taylor expansion can be written as We use similar notations for Ψ j (z) with Ψ j,m−1 (z), ψ (n) j (z 1 , z 2 ) and r p (Ψ j , z).
Then, the following estimate holds for |n| ≤ N + c (c ∈ N is a constant) Proof. Equation (14) implies According to Lemma 3, we have Taking into account that V −1 ,k = O (N m ), we get the following estimate where and This completes the proof in view of the two-point Taylor expansions for functions Φ j and Ψ j at the points −ı and ı.
Lemma 5 Let f (q+2m+1) ∈ AC[−1, 1] for some q, m ≥ 0 and Then, the following estimates hold for p ∈ Z and where Proof. Lemma 3 , with v = 2m + 1 and Equation (14), implies where with . where Hence, which leads to the following estimate for T 1 according to the definition of the binomial coefficients p k for all p ∈ Z (see [123]) Similarly, Equations (32) and (33), along with (30), yield which completes the proof of (27). Relation F −N −p = F N +p completes the proof of (28).

The Pointwise Convergence
The main results of this section are Theorems 3 and 4. They explore the pointwise convergence of the QP-approximations and reveal the exact constants of the corresponding asymptotic errors.

The application of the Abel transformation leads to the expansion
Reiteration of this transformation up toḿ + 1 times implies We prove that The application of similar transformation to (35) leads to the following expansion r = r 1 + r 2 + r 3 + r 4 , where and Equation (15) shows that In view of the following identity (see [123]) By the same arguments, we get In view of Lemma 4 (with v = m + 2) and Lemma 1, we have It shows that Lemma 3 (with v = 2m + 2) indicates that According to Lemma (2), we obtain It means that Finally, we conclude that We continue with estimates of R 1 , R 2 , R 3 and R 4 . Let us start with R 1 . Equation (15) leads to the following equation and, therefore, Then, Lemma 5 implies The following identity (see [123]) and consequently, In view of Lemma 5, By summarizing, we get Similarly, which completes the proof.

Remark 1 The estimate of Theorem 3 is valid only for a fixed x = ±1 and is not a uniform estimate on [−1, 1] (see Theorem 4).
The next theorem reveals the convergence rate of the QP-approximations exactly at the points x = ±1. Preliminary, we need the special case of Lemma 4 with v = 0.
Lemma 6 Let f (q+m) ∈ AC[−1, 1] for some q, m ≥ 0 and Then, the following estimate holds for |n| ≤ N + c (c ∈ N is a constant) Theorem 4 Let f (q+m) ∈ AC[−1, 1] for some q, m ≥ 0 and Then, the following estimate holds as N → ∞ where µ q,m (f, x) and ν q,m (f, x) are defined by (16) and (38) In view of Lemmas 3 and 6, we have These conclude the proof.

Remark 2 Theorem 4 shows convergence rate
It means that the QP-approximations do not converge at the endpoints for q = 0. Accelerated convergence can be achieved by the polynomial correction approach described in Section 5.
Let us compare the behaviors of the quasi-periodic approximations (Algorithm C) and interpolations (see [94,95]). The QP-approximation is written via Vandermonde matrix (12) of size (2m + 2) × (2m + 2), where (2m + 2) is the number of points outside of [−1, 1] used for extensions. Recall also that m = −1 corresponds to the classical truncated Fourier series (without the extension). Theorem 3 proves the convergence rate O(N −q−m−2 ). It means that for each pair of points participating in a function extension (per point for the left and right extensions), the QP-approximation gains additional O(N −1 ) factor in the convergence rate compared to the Krylov-Lanczos approach.
The QP-interpolation is realized by the following Vandermonde matrix Then, the following estimate holds for |x| < 1 We see that each additional m increases the convergence rate by the factor O(N −1 ). In this sense, the QP-interpolation is twice more effective compared to the QP-approximation. 4 The Convergence in the L 2 -norm Theorem 6 is the main result of this section. It reveals the behavior of the QP-approximation in the L 2 -norm.
Proof. We proceed as in the proof of Theorem 3 and extend f outside of [−1, 1] by f (x) ≡ 0: where c n = f n,m − F n,m , |n| ≤ N, c n = f n,m , |n| > N.
Then, we calculate the L 2 [−1, 1]-norm of the error: where First, we estimate I 1 . Lemma 3 and Remark 6 lead to the estimate By tending N to infinity and replacing the sums by the corresponding integrals, we get lim N →∞ Second, we estimate I 2 . We rewrite it as follows According to Lemma 3 and Remark 6, we obtain By tending N to infinity and replacing the sums by the corresponding integrals, we derive Hence, We similarly estimate I 3 and complete the proof.
Theorem 6 is not valid for q = m = 0. However, the same estimate can be proved for q = m = 0 if f ∈ L 2 [−1, 1]. The proof is identical to the one of Theorem 6 and is omitted.
We see that the convergence rate of the QP-approximation in the L 2norm is identical to the rates of the truncated Fourier series and QP-interpolation (see [93]). The constants of the asymptotic errors are different. Formally, c q,−1 (f ) corresponds to the classical Fourier series. Note that the last negative term in (40) vanishes for m = −1. When m = −1, the negative term reduces the L 2 -constant corresponding to the QP-approximation compared to the classical Fourier series.
Unfortunately, the rate of descent depends on the approximated function. Our experiments also reveal that the constant is decreasing as m is increasing. However, the proof of this observation is still unspecified. Let us show it for the following elementary function   Table 1: Numerical values of c q,m (f ) by the QP-approximation for different q and m. The value m = −1 corresponds to the truncated Fourier series.
for the accelerated convergence of the QP-approximations. If the exact values of f (k) (−1) and f (k) (1), k = 0, 1, 2, . . . are known, then the application of the polynomial correction approach will tremendously improve the convergence. More specifically, consider the following representation of f (see also [95]) similar to the Lanczos representation where and P k,q , Q k,q , k = 0, . . . , q − 1 are 2-periodic polynomials satisfying the following conditions and Let us show how polynomials P k,q (x) and Q k,q (x) can be constructed (see [95]). We put if q is even, and if q is odd. Then, the polynomials can be represented by the following recurrent relations k,q−1 (−1) P q−1,q (x) if q − k is even, and Conditions (43) and (44) assure that the function h in (42) satisfies the required derivative-conditions Then, the approximation of h by the QP-approximation leads to the following quasi-periodic-polynomial (QPP-) approximation The Fourier quasi-periodic coefficients of h can be expressed by means of the corresponding coefficients of the functions f , P k,q and Q k,q . Theorems 3, 4 and 6 can be reformulated for the QPP-approximations by omitting the corresponding derivative-conditions. Let us consider only the analog of Theorem 3 for the QPP-approximations.

Implementation Notes
Practical realization of the QPP-approximations is feasible only by efficient approximation of derivatives and calculation of Fourier quasi-periodic coefficients. In general, derivatives at the endpoints are unknown, especially assuming that parameter q in the polynomial correction can tend to infinity. Approximation of the Fourier quasi-periodic coefficients can be performed either by some quadratures (see [124,125]) if the values of f are known on a grid or via known classical Fourier coefficients on [−1, 1] (see [96]). Let us discuss an approach that leads to QPP-approximations with approximate derivatives with better convergence compared to the ones with the exact derivatives. Similar phenomenon was observed and reported in [86,87] for the Eckhoff method named as auto-correction phenomenon. Let us return to (9), where unknown coefficients c lef t j (f ) and c right j (f ), j = 0, . . . , m should be found from the system (10). In particular, the latest is one of the main reasons of the fast convergence of the QP-approximation compared to the truncated Fourier series. We will exploit it for approximation of the derivatives. Let us rewrite (10) for the function h in (42) as follows Taking into account that we get where and, we used 2-periodicity of polynomials P k,q and Q k,q . Now, following the idea behind (10), we consider the following system H n,m = 0, |n| = N − m − q, . . . , N for determination of unknowns A − k (f ), A + k (f ), k = 0, . . . , q − 1 and c lef t j (f ), c right j (f ), j = 0, . . . , m. One of our future works will be the investigation of the system (47) for its effective solutions. Now, let us numerically investigate its impact on the QPP-approximations for a specific simple function. Let which does not accomplish the required derivative-conditions. Figure 1 shows  Figure 2 corresponds to the QPP-approximations with approximate derivatives from the system (47). The comparison of the figures reveals the essence of the auto-correction phenomenon. Inside the interval of approximation, QPPapproximations with approximate jumps have better accuracies compared to the QPP-approximations with the exact jumps. For m = 2 and q = 1, the maximum of the absolute error is 3, 5 · 10 −7 versus 1, 4 · 10 −6 . For m = 2 and q = 4, the maximum of the absolute error is 2 · 10 −12 versus 1, 5 · 10 −9 . As larger is the value of q as bigger is the difference between approximations via exact and numeric derivatives. It is important, that the QPP-approximations with approximate derivatives have improved accuracy on the entire interval as well. For example, we see 6 · 10 −7 versus 1, 2 · 10 −6 for m = 2 and q = 4. We did not observe it in the case of the Eckhoff method. There, the improvements were observed only away from the singularities x = ±1. However, it is easy to explain for the QPP-approximations as interval [−1, 1] is an inside interval for the extended [−1/σ, 1/σ] interval where the approximations are applied. One of our future works should be theoretical investigation of the auto-correction phenomenon for the QPPapproximations.

Conclusion
The paper continued investigations of the quasi-periodic (QP-) approximations initiated in [96]. The main goal was a detailed investigation of Algorithm C proposed in [96], which led to the explicit realization of the approximations through the inverse of a Vandermonde matrix. We explored the convergence of the approximations in different frameworks and derived exact constants for the corresponding asymptotic errors.
Theorem 3 revealed the pointwise behavior of the approximations away from the endpoints. It showed convergence rate O(N −q−m−2 ) as N → ∞, while for the truncated Fourier series (see [86], Theorem 2.4), the convergence rate was O(N −q−1 ) which formally corresponded to the estimate of Theorem 3 with m = −1. We observed improvement in the convergence rate by factor O(N m+1 ), although with different smoothness requirements and derivative-conditions. Theorem 4 estimated the convergence rate of the QP-approximations at x = ±1. It showed O(N −q ) as N → ∞. Hence, the QP-approximations diverged for q = 0 at x = ±1. Theorem 6 explored the convergence in the framework of L 2 [−1, 1]-norm. It showed identical convergence rate O(N −q− 1 2 ) as for the truncated Fourier series. The benefit of the QP-approximation compared to the truncated Fourier series was in smaller constant c q,m (f ), m ≥ 0 compared to c q,−1 (f ) which was based on experimental evidence without the theoretical proof. Additional convergence acceleration was achieved via application of the well-known polynomial correction method (QPP-approximations). We showed the system of linear equations (see (47)) for approximation of the coefficients c lef t j (f ) and c right j (f ) in (46) as well as for determination of derivatives A − k (f ) and A + k (f ) in (42). The approach led to the improved convergence of the QPP-approximations with approximated derivatives compared to the ones that exploited the exact derivatives. Similar improvement was detected for the Eckhoff method and was named as auto-correction phenomenon. In case of the QPP-approximations, the improvement was noticed both on the entire interval and much severe away from the endpoint-singularities.