Convergence of Ces\'{a}ro means with varying parameters of Walsh-Fourier series

Anas Ahmad Abu Joudeh; Gyorgy Gat;


In 2007 Akhobadze [1] (see also [2]) introduced the notion of Ces\`{a}ro means of Fourier series with variable parameters. In the present paper we prove the almost everywhere convergence of the the Ces\`{a}ro $ (C , \alpha_{n})$ means of integrable functions $\sigma _{n}^{\alpha_{n}} f \to f$, where $\mathbb{N}_{\alpha, K}\ni n\to \infty$ for $ f \in L^{1}(I) $, where $I$ is the Walsh group for every sequence $\alpha = (\alpha_n)$, $ 0< \alpha_{n}< 1 $. This theorem for constant sequences $\alpha$ that is, $ \alpha \equiv \alpha_1 $ was proved by Fine [3].

Vol. 19 (2018), No. 1, pp. 303-317

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