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

## DOI:

https://doi.org/10.20535/1810-0546.2015.4.54724## Keywords:

Divergent series, Linear methods of summation, Fourier series, Poisson–Abel method of summation, Poisson–Abel kernel, Normalized basic B-splines## 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.

**R****esults****. **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.

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