Method of Summation of Fourier Series with \[\sigma _{k}(r,\alpha )\] Factors

Abstract

Background. The method of Poisson-Abel type of summation of Fourier series, namely, the method of summation with \[\sigma _{k}(r,\alpha )\] factors is considered in this paper.

Objective. Investigation of method of summation of Fourier series with \[\sigma _{k}(r,\alpha )\] factors.

Methods. We apply the analysis of Poisson-Abel method of summation of such series for investigation of method of summation of Fourier series with \[\sigma _{k}(r,\alpha )\] factors.

Results. It is proved in this paper that application of method of summation with \[\sigma _{k}(r,\alpha )\] factors of Fourier series of periodical function f(t) derives to the convolution of this function with kernels \[De(r,\alpha ,t)\] if the parameter r is integer, these kernels become polynomial normalized basic B-splines of order  ( Also it is proved that for \[\alpha \rightarrow 0\] the method of summation with \[\sigma _{k}(r,\alpha )\] multipliers is F-effective.

Conclusions. We prove that kernels \[De(r,\alpha ,t)\] may be considered as trigonometric representation of normalized basic B-splines of order r – 1 (r = 1, 2, ...)  (and the factors \[\sigma _{k}(r,\alpha )\] are Fourier coefficients of these splines. Other types of finite functions with given properties also may be used as kernels \[De(r,\alpha ,t);\] factors of summation for these kernels are the Fourier coefficients of these kernels. Method of summation of trigonometric series with \[\sigma _{k}(r,\alpha )\] factors may be  applied for summation of trigonometric divergent series with coefficients \[a_{k},b_{k}\]  which have the order of increasing \[O(k^{\beta }), -1< \beta < \infty .\] One can impose the needed properties of smoothness to “generalized” sums of these series.