Behavior of sequences of partial Fourier sums of continuous functions near the points of their divergence
Let s be a point of accmulation of the sequence \(\{S_ n(f;x_ 0)\}\) of partial sums of a function \(f\in C_{2\pi}\) at a point \(x_ 0\), \(s\neq f(x_ 0)\), and let \(\Delta_ s\) be the open interval with end points s and \(2f(x_ 0)-s\). Let \(S_{nk}(f;x_ 0)\to s\) as \(k\to \infty\). It is proved that for every \(c\in \Delta_ s\) there exist two sequences \(\{x_ k\!^{(1)}\}\) and \(\{x_ k\!^{(2)}\}\) convergent to \(x_ 0\) from the left and two sequences \(\{x_ k\!^{(3)}\}\) and \(\{x_ k\!^{(4)}\}\) convergent to \(x_ 0\) from the right such that \(S_{nk}(f,x_ k\!^{(i)})\to s\) for \(i=1,2\) and \(S_{nk}(f;x_ k\!^{(i)})\to 2f(x_ 0)-s\) for \(i=3,4\).
scientific article; zbMATH DE number 3846167