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  • Author or Editor: György Gát x
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Summary  

It is known that the Fejr means - with respect to the character system of the Walsh, and bounded Vilenkin groups - of an integrable function converge to the function a.e. In this work we discuss analogous problems on the complete product of finite, not necessarily Abelian groups with respect to the character system for functions that are constant on the conjugacy classes. We find that the nonabelian case differs from the commutative case. We prove the a.e. convergence of the (C,1) means of the Fourier series of square integrable functions. We also prove the existence  of a function fL q  for some q >1 such that sup |σn f | = + ∞ a.e. This is a sharp contrast between the Abelian and the nonabelian cases.

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Abstract

The aim of this paper is to prove the a.e. convergence of sequences of the Fejér means of the Walsh–Fourier series of bivariate integrable functions. That is, let such that a j (n+1)≧δsup kn a j (n) (j=1,2, n∊ℕ) for some δ>0 and a 1(+∞)=a 2(+∞)=+∞. Then for each integrable function fL 1(I 2) we have the a.e. relation . It will be a straightforward and easy consequence of this result the cone restricted a.e. convergence of the two-dimensional Walsh–Fejér means of integrable functions which was proved earlier by the author and Weisz [3,8].

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The maximal Orlicz spaces such that the mixed logarithmic means of multiple Walsh-Fourier series for the functions from these spaces converge in measure and in norm are found.

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The main aim of this paper is to prove that the nonnegativity of the Riesz’s logarithmic kernels with respect to the Walsh– Kaczmarz system fails to hold.

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Abstract

In 1975 C. F. Chen and C. H. Hsiao established a new procedure to solve initial value problems of systems of linear differential equations with constant coefficients by Walsh polynomials approach. However, they did not deal with the analysis of the proposed numerical solution. In a previous article we study this procedure in case of one equation with the techniques that the theory of dyadic harmonic analysis provides us. In this paper we extend these results through the introduction of a new procedure to solve initial value problems of differential equations with not necessarily constant coefficients.

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The main aim of this paper is to prove that the maximal operator of Marcinkiewicz-Fejér means of double Fourier series with respect to the Walsh-Kaczmarz system is bounded from the dyadic Hardy-Lorentz space H pq into Lorentz space L pq for every p > 2/3 and 0 < q ≦ ∞. As a consequence, we obtain the a.e. convergence of Marcinkiewicz-Fejér means of double Fourier series with respect to the Walsh-Kaczmarz system. That is, σ n ( f, x 1 , x 2 ) → ( x 1 , x 2 ) a.e. as n → ∞.

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