Method of improvement of convergence Fourier series and interpoliation polynomials in orthogonal functions

TL;DR

This paper introduces methods to enhance the convergence of Fourier series and interpolation polynomials using orthogonal function systems, achieving more accurate and uniformly convergent series and reducing interpolation errors.

Contribution

It proposes novel techniques for improving convergence of Fourier series and interpolation polynomials with orthogonal functions, including the concept of phantom nodes.

Findings

Methods demonstrate high efficiency in test cases

Achieve uniformly convergent Fourier series for smooth functions

Significant reduction in interpolation errors

Abstract

There is proposed a method for improving the convergence of Fourier series by function systems, orthogonal at the segment, the application of which allows for smooth functions to receive uniformly convergent series. There is also proposed the method of phantom nodes improving the convergence of interpolation polynomials on systems of orthogonal functions, the application of which in many cases can significantly reduce the interpolation errors of these polynomials. The results of calculations are given at test cases using the proposed methods for trigonometric Fourier series; these calculations illustrate the high efficiency of these methods. Undoubtedly, the proposed method of phantom knots requires further theoretical studies.