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On almost even arithmetical functions via orthonormal systems on Vilenkin groups

100%

Gát G.

Acta Arithmetica

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1991-1992

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tom 60

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nr 2

105-123

2

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

100%

Gát G.

Studia Mathematica

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2001

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tom 144

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nr 2

101-120

EN

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.

3

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

100%

Gát G.

Studia Mathematica

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1998

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tom 130

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nr 2

135-148

EN

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 $σ*f ≔ sup_n |σ_n f|.$ 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.

4

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

63%

Blahota I., Gát G.

Studia Mathematica

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2009

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tom 191

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nr 3

215-222

EN

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.

Ograniczanie wyników

3 Studia Mathematica

1 Acta Arithmetica

4 Gát G.

1 Blahota I.

1 2009

1 2001

1 1998

1 1991

On almost even arithmetical functions via orthonormal systems on Vilenkin groups

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

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

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