Almost everywhere summability of Fourier series with indicating the set of convergence

Abstract

The following problem is studied in this paper: Which multipliers $\{{\lambda }_{k,n}\}$ ensure the convergence, as $n\to \infty$, of the linear means of the Fourier series of functions $f\in L_1[-\pi ,\pi ]$ $$\sum _{k=-\infty }^{\infty }{\lambda }_{k,n}{\hat{f}}_k{e}^{ikx},$$ where ${\hat{f}}_k$ is the $k$-th Fourier coefficient, at a point at which the derivative of the function ${\int }_0^{x}f$ exists. A criterion for the convergence of the $(C,1)$-means (${\lambda }_{k,n}=(1-\frac{|k|}{n+1}{)}_{+}$) is found, while in the general case ${\lambda }_{k,n}=\varphi (\frac{k}{n+1})$ a sufficient condition is derived for the convergence at all such points (that is, almost everywhere). The answer is given in terms of the belonging of $\varphi (x)$ and $x{\varphi }^{\prime }(x)$ to the Wiener algebra of absolutely convergent Fourier integrals. The obtained results are supplemented by some examples.

19 pages, the paper is in Russian with English abstract