Almost Everywhere Summability of Fourier Series with Indication of the Set of Convergence

In this paper, the following problem is studied. For what multipliers $\{\lambda_{k,n}\}$ do the linear means of the Fourier series of functions $f\in L_1[-\pi,\pi]$,
$$ \sum_{k=-\infty}^\infty \lambda_{k,n}\widehat{f}_k e^{ikx}, \qquad \text{where $\widehat{f}_k$ is the $k$th Fourier coefficient}, $$
converge as $n\to \infty$ at all points at which the derivative of the function $\int_0^x f$ exists? In the case $\lambda_{k,n}=(1-|k|/(n+1))_+$, a criterion of the convergence of the $(C,1)$-means and, in the general case $\lambda_{k,n}=\phi(k/(n+1))$, a sufficient condition of the convergence at all such points (i.e., almost everywhere) are obtained. In the general case, the answer is given in terms of whether $\phi(x)$ and $x\phi'(x)$ belong to the Wiener algebra of absolutely convergent Fourier integrals. New examples are given.