On estimates of partial sums of Fourier series of functions of bounded variation (Q1589157)

Let \(\{n_j\}\) be the sequence of all natural numbers \(k\) which satisfy the conditions \(3^p\leq k\leq 3^p+2^p,\) \(p=0,1,2,\dots .\) Then for any periodic function of bounded variation with the Fourier series \(a_0/2+\sum\limits_{m=1}^\infty (a_m\cos mx+b_m\sin mx)\) there holds \[ \sum_{j=1}^\infty \Biggl|\sum_{m=n_j}^ {n_{j+1}-1} (a_m\cos mx+b_m\sin mx) \Biggr|\leq \text{Const}\times\text{total variation of } f. \tag{1} \] This is the main result of the paper. Along with the assertion that there are functions for which (1) does not hold provided \(\lim_{j\to\infty} n_{j+1}/n_j=1,\) this is a certain refinement of an earlier result of the second author.