Summability of trigonometric Fourier series at $d$-points and a generalization of the Abel–Poisson method

We study the convergence of linear means of the Fourier series $\sum_{k=-\infty}^{+\infty}\!\lambda_{k,\varepsilon}\hat{f}_ke^{ikx}$ of a function $f\in L_1[-\pi,\pi]$ to $f(x)$ as $\varepsilon\searrow0$ at all points at which the derivative $\bigl(\int_0^xf(t)\,dt\bigr)'$ exists (i. e. at the $d$-points). Sufficient conditions for the convergence are stated in terms of the factors $\{\lambda_{k,\varepsilon}\}$ and, in the case of $\lambda_{k,\varepsilon}=\varphi(\varepsilon k)$, in terms of the condition that the functions $\varphi$ and $x\varphi'(x)$ belong to the Wiener algebra $A(\mathbb R)$. We also study a new problem concerning the convergence of means of the Abel–Poisson type, $\sum_{k=-\infty}^\infty r^{\psi(|k|)}\hat{f}_ke^{ikx}$, as $r\nearrow1$ depending on the growth of the function $\psi\nearrow+\infty$ on the semi-axis. It turns out that $\psi$ cannot differ substantially from a power-law function.