On an Over-Convergence Phenomenon for Fourier series. Basic Approach

Abstract

This paper is devoted to the acceleration of the convergence of the partial sums of the classical Fourier series for the sufficiently smooth functions. Some universal and adaptive algorithms are constructed and studied. It is shown that the use of a finite number of Fourier coefficients makes it possible exact approximation of a given function from an infinite-dimensional set of quasi-polynomials. In this sense, we call the corresponding essentially nonlinear algorithms as over-convergent. The proposed algorithms are implemented using Wolfram Mathematica system. Numerical results demonstrate their effectiveness.

A correction to the article "On an Over-Convergence Phenomenon for Fourier Series. Basic Approach"

“…In the item 4.3.3 of my work [1], the last formula of Theorem 3 contains errors. Below is the main part of the item, including the revised proof of Theorem 1, which corresponds to Theorem 3 of [1]. The remaining numbering from [1] is preserved.…”

“…Below is the main part of the item, including the revised proof of Theorem 1, which corresponds to Theorem 3 of [1]. The remaining numbering from [1] is preserved.…”

On some quasi-periodic approximations

“…For the realization of the approximation S N,m (G, x), the Fourier quasiperiodic coefficients of G can be computed from (15). Since the exact values of jumps A − k (f ) and A + k (f ) are unknown, the corresponding approximations can be derived from the following system of linear equations G n,m = 0, |n| = N − q + 1, .…”

“…Different approaches are known for convergence acceleration of the truncated Fourier series or trigonometric interpolation for non-periodic smooth functions. We refer to [4][5][6][7][8][9][10][11][12][13][14][15][16][17] with references therein for a detailed review of different methods.…”

Acceleration of Convergence of Fourier Series Using the Phenomenon of Over-Convergence

“…Over the past three decades, in numerous works devoted to the scheme of the KE-method, not only polynomials were used as a function g when g s = 0 for all s. Here we will not dwell even on a brief review since, as far as we know, the approach proposed below has not been used before. Those who wish can fill this gap, for example, in [2][3][4][5][6][7][8][9][10][11] and references therein.…”