Vilenkin Groups Research Articles

On the absolute convergence of Vilenkin-Fourier series

In extending Bernstein's theorem on the absolute convergence of the Fourier series to Vilenkin groups, Quek and Yap ( J. Math. Anal. Appl. 41 (1980) 1–14) assume that the group G has the boundedness property (P). The main purpose of the present paper is to provide a proof to the result of Quek and Yap without the condition of the boundedness property (P) on G.

Orthonormal systems on Vilenkin groups

Orthonormal systems on Vilenkin groups

Fourier transforms of Dini‐Lipschitz functions on Vilenkin groups

In [4] we proved some theorems on the Fourier Transforms of functions satisfying conditions related to the Dini‐Lipschitz conditions on the n‐dimensional Euclidean space Rn and the torus group Tn. In this paper we extend those theorems for functions with Fourier series on Vilenkin groups.

On almost even arithmetical functions via orthonormal systems on Vilenkin groups

On almost even arithmetical functions via orthonormal systems on Vilenkin groups

WeightedH p multipliers on locally compact Vilenkin groups

LetG be a locally compact Vilenkin group. We give a maximal function characterization of the weightedHp spaces overG, and give a Hormander type multiplier theorem for these spaces.

Sets of uniqueness and closed subgroups in Vilenkin groups

Мы доказываем, что зам кнутые нулевой меры п одгруппы группы Виленкина (огр аниченной или нет) являются множ ествами единственно сти. Новые идеи, привлеченные дл я наших рассмотрений: примен ение двойственности Понтрягина к вопросам единственн ости, изучение общих групп Виленкина, весьма общ ее упорядочение характ еров.

Multipliers on weighted Hardy spaces over locally compact Vilenkin groups, I

AbstractLet G denote a locally compact Vilenkin group with dual group Γ. We give sufficient conditions for a function ϕ ∈ L∞ (Γ) to be a multiplier from the power-weighted Hardy space to itself or the corresponding power-weighted Lebesgue space

Multipliers on weighted 𝐿_{𝑝}-spaces over locally compact Vilenkin groups

Let G G be a locally compact Vilenkin group. We prove a multiplier theorem for power-weighted L p ( G ) {L_p}(G) -spaces which extends an earlier result of this kind, due to Onneweer.

Smoothness conditions and integrability theorems on bounded Vilenkin groups

AbstractVarious 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.

Summation methods and uniqueness in Vilenkin groups

It is shown that the σ \sigma -compact U U -sets for two different summation methods on Vilenkin groups are the same.

Multipliers on weighted Hardy spaces over certain totally disconnected groups

In this note, we consider the multipliers on weighted H1 spaces over totally disconnected locally compact abelian groups with a suitable sequence of open compact subgroups (Vilenkin groups). We first show an (H1, L1) multiplier result from which Onneweer′s theorem follows. We also give an (H1, H1) multiplier result under a condition of Baernstein‐Sawyer type.

Sets of uniqueness in compact, 0-dimensional metric groups

An investigation is made of sets of uniqueness in a compact 0 0 -dimensional space. Such sets are defined by pointwise convergence of sequences of functions that generalize partial sums of trigonometric series on Vilenkin groups. Several analogs of classical uniqueness theorems are proved, including a version of N. Bary’s theorem on countable unions of closed sets of uniqueness.

Generalized Lipschitz Spaces on Vilenkin Groups

It is shown that, for certain choices of the defining indices, the generalized Lipschitz spaces on Vilenkin groups are included in certain Figa-Talamanca spaces Ap, and that the Fourier series of functions in the latter spaces converge uniformly. This result includes an extension of the classical Dini test, and endpoint versions of Bernstein's theorem on absolute convergence of Fourier series.

Compact Sets of Divergence for Continuous Functions on a Vilenkin Group

Compact Sets of Divergence for Continuous Functions on a Vilenkin Group

Compact sets of divergence for continuous functions on a Vilenkin group

Let G G be a Vilenkin group. Let E ⊂ G E \subset G be closed with Haar measure zero. We show there is a continuous function whose Vilenkin-Fourier series diverges at every point in E E .

On the modulus of continuity with respect to functions defined on Vilenkin groups

ha this paper we generalize a theorem of Rubinstein [2]. By the usual definition the modulus of continuity of a function defined on a Vilm,&in group (G,,) can be represented by a sequence of real numbers. Rubinstein [2] characterized the sequences which turn to be the modulus of continuity of a function in C(G,,), L~(G~), LZ(G=). Now we prove his conjecture, namely the theorem is true for LP(Gm) (t ~ p < ~) too.

Multipliers of Lipschitz spaces on a certain class of groups

Zygmund [lo], Mizuhara [7], and Anderson [l] have studied various multiplier problems of Lipschitz spaces defined on the circle group T. In [8,9], the authors have characterized the multipliers of Lipschitz spaces defined on a Vilenkin group G (i.e., G is an infinite compact O-dimensional metric Abelian group). In this paper we define a certain class S of metric locally compact Abelian groups, and study the multipliers from one Lipschitz space Lip(a,p; G) to another Lipschitz space Lip@, q; G) for G in .Y. The class .Y contains the classical groups T” and R”, as well as many other interesting groups. Our main result is a characterization of the multipliers from Lip(a,p; G) to Lip@‘, q; G), as restrictions of multipliers from Lp(G) to L9(G), for GE ,Y, 0 < a </3 < 1, 1 <p < co, and l<q<co.

Multipliers from one Lipschitz space to another

Let G be a locally compact Abelian group with Haar measure 1. For 1 < r < co, L,(G) will denote the usual Lebesgue space defined on G with respect to 1. In recent years a number of papers have been devoted to the study of multipliers from L,(G) to a function space defined on G. For example, Burnham and Goldberg [ 1 ] and Goldberg and Seltzer [3] have studied the multipliers from L,(G) to a Segal algebra S(G); Feichtinger [2] has studied the multipliers from L,(G) to a homogeneous Banach space B(G); in [6] we have characterized the space of multipliers from L,(G) to a Lipschitz space Lip(a,p, G) (or lipp G) (or Ll,(a,p; G)), where G is a metrizable locally compact Abelian group (with additional hypothesis imposed on G in some of our results). In this note we present two results (see Theorems 1 and 2) on the multipliers from one Lipschitz space yip(a,p; G) to another Lipschitz space Uip(/.?, q; G), where G is a Vilenkin group (i.e., G is an infinite, compact, O-dimensional, metrizable, Abelian group). Zygmund [ 111 and Mizuhara [ 51 have obtained similar results for Lipschitz spaces defined on the circle group. For the remainder of this note G will denote a Vilenkin group. We shall need some facts about G; the relevant facts, notation and terminology can be found in the first two pages of our paper [7]. The following definitions are slight extensions of Definitions 2.2 and 2.3 in

Factorization of Lipschitz functions and absolute convergence of Vilenkin-Fourier series

LetG be a Vilenkin group (i.e., an infinite, compact, metrizable, zero-dimensional Abelian group). Our main result is a factorization theorem for functions in the Lipschitz spaces\(\mathfrak{L}\mathfrak{i}\mathfrak{p}\) (α,p; G). As colloraries of this theorem, we obtain (i) an extension of a factorization theorem ofY. Uno; (ii) a convolution formula which says that\(\mathfrak{L}\mathfrak{i}\mathfrak{p} (\alpha , r; G) = \mathfrak{L}\mathfrak{i}\mathfrak{p} (\beta , l; G)*\mathfrak{L}\mathfrak{i}\mathfrak{p} (\alpha - \beta , r; G)\) for 0<β<α<∞ and 1≤r≤∞; and (iii) an analogue, valid for allG, of a classical theorem ofHardy andLittlewood. We also present several results on absolute convergence of Fourier series defined onG, extending a theorem ofC. W. Onneweer and four results ofN. Ja. Vilenkin andA. I. Rubinshtein. The fourVilenkin-Rubinshtein results are analogues of classical theorems due, respectively, toO. Szasz, S. B. Bernshtein, A. Zygmund, andG. G. Lorentz.