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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...

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 group decompositions of bounded cosine sequences

Wojciech Chojnacki (2007)

Studia Mathematica

A two-sided sequence ( c ₙ ) n ∈ ℤ with values in a complex unital Banach algebra is a cosine sequence if it satisfies c n + m + c n - m = 2 c ₙ c ₘ for any n,m ∈ ℤ with c₀ equal to the unity of the algebra. A cosine sequence ( c ₙ ) n ∈ ℤ is bounded if s u p n ∈ ℤ | | c ₙ | | < ∞ . A (bounded) group decomposition for a cosine sequence c = ( c ₙ ) n ∈ ℤ is a representation of c as c ₙ = ( b ⁿ + b - n ) / 2 for every n ∈ ℤ, where b is an invertible element of the algebra (satisfying s u p n ∈ ℤ | | b ⁿ | | < ∞ , respectively). It is known that every bounded cosine sequence possesses a universally defined group decomposition, here referred...

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