On the convergence of negative-order Cesàro means of Fourier and Fourier-Walsh series (Q2090548)

Recently it was proved by \textit{M. G. Grigoryan} and \textit{L. N. Galoyan} [J. Math. Anal. Appl. 434, No. 1, 554--567 (2016; Zbl 1331.42004)] that there exists an increasing subsequence of positive integers \(\Lambda = \{M_{\nu}\}_{\nu =1}^{\infty}\) with density \(1\) such that for any function \(f(x) \in L^{0}[-\pi, \pi]\) and any number \(\varepsilon > 0\), one can define a function \(g\) continuous on \([-\pi, \pi]\) with \(\mu \{f\not = g\}<\varepsilon\) and such that the subsequence \(\sigma_{M_{\nu}}^{\alpha}(g, T)\), \(\nu = 0,1, \dots,\) of Cesáro means of order \(\alpha \in (-1, 0)\) of the trigonometric Fourier series of \(g\) uniformly converges on \([-\pi, \pi]\). In the present paper, it is stated that in the case when \(\alpha <-\frac{1}{2}\), \(\alpha \not = -1, -2, \dots,\) this theorem can be strengthened, namely the corrected function \(g\) will have, in addition, uniformly convergent trigonometric Fourier series. The author notes also that the theorem remains true for the Walsh system.