On the convergence to infinity of Fourier series along dense subsequences of numbers (Q1261261)

In the paper is proven the Theorem. For every increasing sequence of natural numbers \(\{H_ k\}^ \infty_{k=1}\) with \(\lim_{k\to\infty} {H_ k\over k}=+\infty\), there exists \(f\in L(0,2\pi)\) such that for almost every \(x\in (0,2\pi)\) one can find an increasing sequence \(\{n_ k(x)\}^ \infty_{k=1}\) of numbers satisfying the conditions \[ n_ k(x)\leq H_ k,\qquad k=1,2,\dots, \] \[ \lim_{k\to\infty} S_{n_{2k}(x)}(x, f)=+ \infty,\quad \lim_{k\to\infty} S_{n_{2k- 1}(x)}(x,f)= -\infty, \] where \(S_ n(x,f)\) denotes the \(n\)th partial sum of the Fourier series of the function \(f\).