Mean convergence of Fourier series (Q803459)

The author improves the results of \textit{S. A. Telyakovskij} [Mat. Zametki 1, 91-98 (1967; Zbl 0202.063)] dealing with the convergence in the metric L of series \((1)\quad a_ 0/2+\sum a_ k \cos kx\) and \((2)\quad \sum a_ k \sin kx.\) It is proved that if one of the conditions \(\lim_{n\to \infty}a_ n \log n=0\) formulated in the cited paper is substituted by the weaker condition \(\lim_{n\to \infty}\sum^{[n/2]}_{k=1}\frac{1}{k}(| a_{n-k}| +| a_{n+k}|)=0,\) then the series (1) is L convergent again. A similar result is valid for the series (2). Further, conditions for the uniform boundedness in L of the partial sums of (1) and (2) are given.