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On (C,1) summability of integrable functions with respect to the Walsh-Kaczmarz system

G. Gát — 1998

Studia Mathematica

Let G be the Walsh group. For f L 1 ( G ) we prove the a. e. convergence σf → f(n → ∞), where σ n is the nth (C,1) mean of f with respect to the Walsh-Kaczmarz system. Define the maximal operator σ * f s u p n | σ n f | . We prove that σ* is of type (p,p) for all 1 < p ≤ ∞ and of weak type (1,1). Moreover, σ * f 1 c | f | H , where H is the Hardy space on the Walsh group.

On (C,1) summability for Vilenkin-like systems

G. Gát — 2001

Studia Mathematica

We give a common generalization of the Walsh system, Vilenkin system, the character system of the group of 2-adic (m-adic) integers, the product system of normalized coordinate functions for continuous irreducible unitary representations of the coordinate groups of noncommutative Vilenkin groups, the UDMD product systems (defined by F. Schipp) and some other systems. We prove that for integrable functions σₙf → f (n → ∞) a.e., where σₙf is the nth (C,1) mean of f. (For the character system of the...

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