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In this paper we study the a.e. strong summability of the quadratical and rectangular partial sums of the two-dimensional Vilenkin--Fourier series for every two-dimensional functions belonging to Llog L . We also study the exponential uniform strong approximation of Marcinkiewicz type of two-dimensional Vilenkin--Fourier series. [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 article is to investigate the (Hp, Lp)-type inequality for the maximal operators of Riesz and Nörlund logarithmic means of Walsh–Kaczmarz–Fourier series. Moreover, we show that the behaviour of Nörlund logarithmic means is worse than the behaviour of Riesz logarithmic means in our special sense. [ABSTRACT FROM AUTHOR]

In the present paper we prove that for any 0

The main aim of this paper is to prove that the logarithmic means of quadratical partial sums of the double Walsh-Kaczmarz series does not improve the convergence in measure. In other words, we prove that for any Orlicz space, which is not a subspace of L log+ L(I²), the set of the functions the logarithmic means of quadratical partial sums of the double Walsh-Kaczmarz series of which converge in measure is of first Baire category. [ABSTRACT FROM AUTHOR]

For the two-dimensional Walsh system, Gát and Weisz proved the a.e. convergence of Fejér means σ f of integrable functions, where the set of indices is inside a positive cone around the identical function, that is, β ≤ n/ n ≤ β is provided with some fixed parameter β ≥ 1. In this paper we generalize the result of Gát and Weisz. We not only generalize this theorem, but give a necessary and sufficient condition for cone-like sets in order to preserve this convergence property. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to prove that the Nörlund logarithmic means tnkf of one-dimensional Walsh-Kaczmarz-Fourier series is weak type (1,1), and this fact implies that tnkf converges in measure on I for every function f ∈ L(I) and tn,mkf converges in measure on I2 for every function f ∈ Lln+ L(I2): Moreover, the maximal Orlich space such that Nörlund logarithmic means of two-dimensional Walsh-Kaczmarz-Fourier series for the functions from this space converge in two-dimensional measure is found. [ABSTRACT FROM AUTHOR]