On the series of absolute values of blocks of trigonometric series
Let \(\Lambda\) be a strictly increasing sequence of natural numbers, \(1=n_{1}<n_{2}< ... \). To the series \(\sum_{k=0}^{\infty}a_{k}\cos(k x)\) and \(\sum_{k=0}^{\infty}b_{k}\sin(k x)\), with \(a_{k}, b_{k}\in \mathbb{R}\), \(k\in \mathbb{N}\cup \{0\},\) and \(\lim_{k\to \infty}a_{k}=\lim_{k\to \infty}b_{k}=0\), one associates the series \[ F_{\Lambda}(x)=\sum_{j=1}^{\infty}\left |\sum_{k=n_{j}}^{n_{j+1}-1}a_{k}\cos(k x)\right |, \, \, G_{\Lambda}(x)=\sum_{j=1}^{\infty}\left |\sum_{k=n_{j}}^{n_{j+1}-1}b_{k}\sin(k x)\right |, \] where evidently the values of \(F_{\Lambda}(x)\) and \(G_{\Lambda}(x)\) may be \(+\infty\). The main results of the paper can be stated as follows. { Theorem.} Let \(\gamma\in (0, 1)\), \(a_{k}, b_{k}\in \mathbb{R}\), \(k\in \mathbb{N}\cup \{0\},\) with \(\lim_{k\to \infty}a_{k}=\lim_{k\to \infty}b_{k}=0\). (i) If the series \(\sum_{j=1}^{\infty}\sum_{k=n_{j}}^{n_{j+1}-1}|a_{k}-a_{k+1}|(k-n_{j}+1)^{\gamma}\) converges, then the integral \(\int_{0}^{\pi}\frac{1}{x^{\gamma}}\cdot F_{\Lambda}(x)dx \) converges if and only if the series \(\sum_{j=1}^{\infty}|a_{n_{j+1}}|(n_{j+1}-n_{j})^{\gamma}\) converges. (ii) If the series \(\sum_{j=1}^{\infty}\sum_{k=n_{j}}^{n_{j+1}-1}|a_{k}-a_{k+1}|\log(k-n_{j}+2)\) converges, then \(F_{\Lambda}\) is Lebesgue integrable on \([0, \pi]\) if and only if the series \(\sum_{j=1}^{\infty}|a_{n_{j+1}}\log(n_{j+1}-n_{j}+1)\) converges. (iii) If the series \(\sum_{j=1}^{\infty}\sum_{k=n_{j}}^{n_{j+1}-1}|b_{k}-b_{k+1}|(k-n_{j}+1)^{\gamma}\) converges, then the integral \(\int_{0}^{\pi}\frac{1}{x^{\gamma}}\cdot G_{\Lambda}(x)dx \) converges if and only if the series \(\sum_{j=1}^{\infty}|b_{n_{j+1}}|(n_{j+1}-n_{j})^{\gamma}\) converges.
scientific article; zbMATH DE number 6381017