For $2\pi$-periodic functions $f$ that have, on $[-\pi,\pi]$, only the point 0 as a nonsummable singular point, we consider generalized Fourier series depending on an integer-valued function $N(x)$. It is shown that if $|x|^{\alpha(x)}f(x)\in L(-\pi,\pi)$, where $\alpha(x)$ is an even nonnegative function, nonincreasing on $(0,\pi]$, and $\alpha(x)=o(\ln\frac1x)$, $x\to+0$, then under a certain condition on $N(x)$ the generalized Fourier series is almost everywhere summable to $f(x)$ by the Abel method. The estimate $o(\ln\frac1x)$ and the hypothesis on $N(x)$ are, in a certain sense, definitive. Bibliography: 3 titles.