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

## Authors

• Володимир Петрович Денисюк National Aviation University, Ukraine

## 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.

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.

## Author Biography

### Володимир Петрович Денисюк, National Aviation University

Volodymyr P. Denysiuk.

Doctor of physics and mathematics, full professor, head of the Department of Higher and Numerical Mathematics

## References

G.H. Hardy, Divergent Series. Мoscow, USSR: Izdatelstvo Inostrannoyi Literatury, 1951, 504 p. (in Russian).

V.К. Dziadyk, Introduction to Theory of Uniform Functions Approximation by Polynomials. Мoscow, USSR: Nauka, 1977, 512 p. (in Russian).

N.K. Bari, Trigonometric Series. Мoscow, USSR: Gosizdat Fiz.-Mat. Literatury, 1961, 936 p. (in Russian).

V.P. Denisiuk, “Method of improving of convergence for fourier series and interpolating polynomial in orthogonal function”, Naukovi Visti NTTU “КPІ”, no. 4, pp. 31–37, 2013 (in Ukrainian).

C. Lanchos, Applied Analysis. Мoscow, USSR: Gosizdat Fiz.-Mat. Literatury, 1961, 524 p. (in Russian).

V.P. Denisiuk, Fundamental Functions and Trigonometric Splines. Кyiv, Ukraine: PAT “VIPOL”, 2015, 296 p. (in Ukrainian).

Yu.S. Zav’yalov et al., Methods of Spline Functions. Мoscow, USSR: Nauka, 1980, 352 p. (in Russian).

C. de Boor, A Practical Guide to Splines. Мoscow, USSR: Radio i Sviaz, 1985, 304 p. (in Russian).

A. Zygmund, Trigonometric Series, 2 vol. Мoscow, USSR: Мir, 1965. (in Russian).