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In the present paper we give a necessary and sufficient condition on monotone weights, which guaranties the Nörlund means of Walsh-Fourier series converge in L1 norm and CW norm. We also discuss the almost everywhere summability by Nörlund means and we prove that our condition is sufficient for almost everywhere convergence of Nörlund means for all integrable functions. [ABSTRACT FROM AUTHOR]

The presented paper discusses the matrix summability of the Walsh–Fourier series. In particular, we discuss the convergence of matrix transforms in L 1 space and in C W space in terms of modulus of continuity and matrix transform variation. Moreover, we show the sharpness of our result. We also discuss some properties of the maximal operator t ∗ (f) of the matrix transform of the Walsh–Fourier series. As a consequence, we obtain the sufficient condition so that the matrix transforms t n (f) of the Walsh–Fourier series are convergent almost everywhere to the function f. The problems listed above are related to the corresponding Lebesgue constant of the matrix transformations. The paper sets out two-sides estimates for Lebesgue constants. The proven theorems can be used in the case of a variety of summability methods. Specifically, the proven theorems are used in the case of Cesàro means with varying parameters. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to investigate the Walsh-Marcinkiewicz means on the Hardy space H, when 0 < p < 2/3. We define a weighted maximal operator of Walsh-Marcinkiewicz means and establish some of its properties. With its aid we provide a necessary and sufficient condition for convergence of the Walsh-Marcinkiewicz means in terms of modulus of continuity on the Hardy space H, and prove a strong convergence theorem for the Walsh-Marcinkiewicz means. [ABSTRACT FROM AUTHOR]