The method of summation of divergent trigonometric series

TL;DR

This paper introduces a generalized summation method for divergent trigonometric series using $\sigma_k(r,a)$-factors, linking it to convolution with specific kernels, including polynomial B-splines, and demonstrates its effectiveness.

Contribution

It presents a new summation technique for divergent Fourier series, connecting it to convolution with B-spline kernels and proving its effectiveness.

Findings

Summation results in convolution with kernels $De(r,\alpha,t)$.

When $r$ is integer, kernels are polynomial B-splines.

The method is proven to be $F$-effective.

Abstract

The generalized summation of divergent trigonometric series, namely by method of $\sigma_k(r,a)$-factors is considered in this paper. It is proved that such summation of Fourier series of periodical function $f(t)$ results in the convolution of this function with kernels $De(r,\alpha,t)$; if the parameter $r$ is integer, these kernels are polynomial normalized basic $B$-splines of order $r-1$ $(r=1,2,\ldots)$. Also it is proved that the method of summation with $\sigma_k(r,a)$-factors is $F$-effective.