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

Abstract

In recent publications of the author, the phenomenon of over-convergence was discovered, and a spectral method has been presented for accelerating the convergence of truncated Fourier series for smooth functions. On this basis, a certain parametric system that is biorthogonal to the corresponding segment of the Fourier system turned out to be unusually effective. This article reconsiders some approaches and makes some adjustments to previous publications. As a result, two improved schemes for the recovery of a function based on a finite set of its Fourier coefficients are proposed. Numerical experiments confirm a significant increase in the efficiency of corresponding algorithms in typical classes of smooth functions. In conclusion, some prospects for the development and generalization of the above approaches are discussed.

Exploring the Potential of Mixed Fourier Series in Signal Processing Applications Using One-Dimensional Smooth Closed-Form Functions with Compact Support: A Comprehensive Tutorial

“…For instance, the works of A. Nersessian and A. Poghosyan addressed the main issue of some alternatives to the quasi-Bernoulli series in [67,[87][88][89][90][91][92], such as the quasi-polynomial series, the Fourier-Pade series, the trigonometric interpolations series, and the quasi-polynomial Pade series. Similarly, A. Nersessian studied a framework based on a biorthogonal system and adaptive algorithms with a strong potential for accelerating the convergence of Fourier series due to an over-convergence phenomenon [93][94][95][96]. Furthermore, some simplifications and applications of Eckhoff algorithm have been studied by A.…”

“…Furthermore, some simplifications and applications of Eckhoff algorithm have been studied by A. Poghosyan et al in [83][84][85][86]97], such as its application to two-dimensional functions, the simplification of the minimization problem, and the study of trigonometric interpolations series. Finally, several researchers around the world, who are not fully discussed in this introduction due to space limitations, have also contributed to this technique [82,[98][99][100][101].…”