Abstract. The main aim of this paper is to prove that the maximal operator σ * 0 := sup n |σ n,n | of the Fejér means of the double Vilenkin-Fourier series is not bounded from the Hardy space H 1/2 to the space weak-L 1/2 .Let N + denote the set of positive integers, N := N + ∪ {0}. Let m := (m 0 , m 1 , . . .) be a sequence of positive integers not less than 2. Denote by Z m k := {0, 1, . . . , m k − 1} the additive group of integers modulo m k . Define the group G m as the complete direct product of the groups Z m j , with the product of the discrete topologies of Z m j 's. The direct product µ of the measuresis the Haar measure on G m with µ(G m ) = 1. If the sequence m is bounded, then G m is called a bounded Vilenkin group, otherwise it is an unbounded Vilenkin group.

Abstract. As main result we prove strong convergence theorems of Vilenkin-Fejér means when 0 < p ≤ 1/2.2010 Mathematics Subject Classification. 42C10.

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The (Nörlund) logarithmic means of the Fourier series is:In general, the Fejér (C, 1) means have better properties than the logarithmic ones. We compare them and show that in the case of some unbounded Vilenkin systems the situation changes.

In this article we discuss the behaviour of Θ-means of quadratical partial sums of double Walsh series of a function in L p (G 2) (1 p ∞). In case p = ∞ by L p (G 2) we mean C , the collection of continuous functions on G 2. We present the rate of the approximation by Θ-means, in particular, in Lip(α, p), where α > 0 and 1 p ∞. Our main theorem generalizes two result of Nagy on Nörlund means and weighted means of the cubical partial sums of double Walsh-Fourier series [15, 16]. Specifically, we give the twodimensional analogue of the two results of Móricz, Siddiqi on Nörlund means [14] and Móricz, Rhoades on weighted means [12].