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

• Autorzy: G. Gát
• Języki publikacji: EN

Abstrakty

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.

Kategorie tematyczne

• 43A75: Analysis on specific compact groups
• 42C15: General harmonic expansions, frames
• 40G05: Ces\`aro, Euler, N\"orlund and Hausdorff methods
• 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2001
• Tom: 144
• Numer: 2
• Strony: 101-120

Daty

wydano

2001

Twórcy

autor

G. Gát

• Department of Mathematics, Bessenyei College, P.O. Box 166, H-4400 Nyíregyháza, Hungary

Identyfikatory

DOI

10.4064/sm144-2-1