The method of summation of divergent trigonometric series

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 $r-1$ $(r=1,2,\dots )$. Also it is proved that the method of summation with ${\sigma }_k(r,a)$-factors is $F$-effective.