Convergence of trigonometric and Walsh-Fourier series

Abstract: In this paper we present some results on convergence and summability of one- and multi-dimensional trigonometric and Walsh-Fourier series. The Fej{é}r and Cesaro summability methods are investigated. We will prove that the maximal operator of the summability means is bounded from the corresponding classical or martingale Hardy space $H_p$ to $L_p$ for some $p>p_0$. For $p=1$ we obtain a weak type inequality by interpolation, which ensures the almost everywhere convergence of the summability means.

Keywords: Martingale and classical Hardy spaces, $p$-atom, atomic decomposition, interpolation, Walsh functions, trigonometric functions, Fej{é}r summability, $(C,\alpha)$ summability

Classification (MSC2000): 2B08; 42C10, 42B30, 42B25, 60G42

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