Book-Review-Dynamical Systems-III

Acta Mathematica Hungarica, 2010

We extend the Dezern-Waterman version of the Lebesgue test for convergence of Vilenkin-Fourier series to the case of unbounded Vilenkin groups.

Acta Mathematica Hungarica, 2017

We study approximation by rectangular partial sums of double Fourier series on unbounded Vilenkin groups in the spaces C and L1. From these results we obtain criterions of the uniform convergence and L-convergence of double Vilenkin-Fourier series. We also prove that these results are sharp.

1999

In this paper we prove that for a function f # L p (G m) where 1<p the Feje r means _ n f converge to f almost everywhere with respect to the character system of any (bounded or not) Vilenkin group G m .

Journal of Fourier Analysis and Applications, 2009

We study norm convergence and summability of Fourier series in the setting of reduced twisted group C * -algebras of discrete groups. For amenable groups, Følner nets give the key to Fejér summation. We show that Abel-Poisson summation holds for a large class of groups, including e.g. all Coxeter groups and all Gromov hyperbolic groups. As a tool in our presentation, we introduce notions of polynomial and subexponential H-growth for countable groups w.r.t. proper scale functions, usually chosen as length functions. These coincide with the classical notions of growth in the case of amenable groups.

Arkiv för matematik, 1982

Acta Mathematica Sinica, English Series, 2007

It is a highly celebrated problem in dyadic harmonic analysis the pointwise convergence of the Fejér (or (C, 1)) means of functions on unbounded Vilenkin groups. There are several papers of the author of this paper concerning this. That is, we know the a.e. convergence σ n f → f (n → ∞) for functions f ∈ L p , where p > 1

Journal of the Australian Mathematical Society, 1988

Various criteria, in terms of forward differences and related operations on coefficients, are shown to imply that certain series on bounded Vilenkin groups represent integrable functions. These results include analogues of known integrability theorems for trigonometric series. The method of proof is to pass from the given series to a derived series, and to deduce the integrability of the original series from smoothness properties of the latter.

Analysis Mathematica, 1996

It is well known that the partial sums of the Vilenkin-Fourier series of any f ∈ L p , 1 < p < ∞, converge to f in norm. For every 1 ≤ p ≤ ∞ the S M n operators for any f ∈ L p also converge to f in norm. In this work we study the above affirmations in similar, not necessarily Abelian, totally disconnected groups and the product system of normed coordinate functions for continuous irreducible unitary representations of coordinate groups. Finally we prove the L p-norm convergence of Fejér means for 1 ≤ p ≤ ∞ in the case of bounded groups.

1995

It is known that the Fejér means-with respect to Vilenkin systemsof an integrable function on bounded Vilenkin groups converge to the function a.e. In this work we prove the above on similar, not necessarily Abelian, totally disconnected groups with respect to the product system of normed coordinate functions of continuous irreducible unitary representations of the coordinate groups.

1970

Acta Mathematica Hungarica, 1986

In this paper we deal with the connection (in different spaces) among the Vilenkin--Fourier sums, the modulus of continuity and the Lebesgueconstants (with respect to the Vilenkin-system). We give two sided estimates for an expression containing these quantities. The corresponding problem for the trigonometric system was considered by Lebesgue [7] and Oskolkov . . Let m:=(mk, kEN) (N={0, 1, ...}) be a sequence of natural numbers, whose terms are not less than 2. Denote by Z,,~ (kEN) the discrete cyclic group of order rnk, and define G,, as the direct product of Z~'s (endowed vr the product topology and measure). Gm is a compact Abelian group with the normalized Haar measure #. The elements of G~ are of the form x= (x0, xa, ..., xk .... ) (O~Xk-<mk, k, xkEN ). Further we need the following subsets of Gm:

Russian Mathematics, 2016

Some results concerning the summability of Fourier series of continuous 2π-periodic functions are generalized for the case of almost-periodic functions defined on locally compact Abelian groups.

Banach Journal of Mathematical Analysis, 2018

In this paper we prove that, in the case of some unbounded Vilenkin groups, the Riesz logarithmic means converges in the norm of the spaces X(G) for every f ∈ X(G), where by X(G) we denote either the class of continuous functions with supremum norm or the class of integrable functions.

1991

Proceedings of the American Mathematical Society, 1969

Only real-analytic functions operate in the Fourier algebra of any compact group that has an infinite abelian subgroup. This extends the theorems of Helson, Kahane, Katznelson, and Rudin [4] which apply to the algebra of absolutely convergent Fourier series on compact abelian groups. The Fourier algebra of a locally compact group has been studied by H. Mirkil [ó], W. F. Stinespring [9], R. A. Mayer [5], C. Herz [3], and most thoroughly by P. Eymard [l]. We will state here the relevant definitions and facts, and prove that the restriction of the Fourier algebra to a closed subgroup is the Fourier algebra of the subgroup, and use this to lift up the theorem on operating functions.