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

100%

Gát G.

Studia Mathematica

2001

tom 144

nr 2

101-120

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 group of m-adic integers, this proves a more than 20 years old conjecture of M. H. Taibleson [24, p. 114].) Define the maximal operator σ*f : = supₙ|σₙf|. 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.

Almost everywhere convergence of Marcinkiewicz means of Fourier series on the group of 2-adic integers

63%

Blahota I. , Gát G.

Studia Mathematica

2009

tom 191

nr 3

215-222

We prove the almost everywhere convergence of the Marcinkiewicz means of integrable functions σₙf → f for every f ∈ L¹(I²), where I is the group of 2-adic integers.