Zero-dimensional Group Research Articles

Tight wavelet frames on zero-dimensional groups. Construction and approximation

We present a method for constructing tight wavelet frames on locally compact zero-dimensional groups. When constructing frames, we do not use the principle of unitary extension. We also consider the approximate properties of the resulting frames for functions from the Sobolev space with logarithmic weight.

The Bestvina–Edwards theorem and the Hilbert–Smith conjecture

We prove a number of results surrounding the Borsuk–Ulam-type conjecture of Baum, Dabrowski, and Hajac (BDH, for short), which states that given a free action of a compact group G on a compact space X, there are no G-equivariant maps X∗G→X (with ∗ denoting the topological join). Mainly, we prove the BDH conjecture for locally trivial principal G-bundles. The proof relies on the nonexistence of G-equivariant maps G∗(n+1)→G∗n, which in turn is a strengthening of an unpublished result of Bestvina and Edwards. Moreover, we show that the BDH conjecture partially settles a conjecture of Ageev which implies the weak version of the Hilbert–Smith conjecture stating that no infinite compact zero-dimensional group can act freely on a manifold so that the orbit space is finite-dimensional.

Reconstruction of a Generalized Fourier Series from Its Sum on a Compact Zero-Dimensional Group in the Non-Abelian Case

Reconstruction of a Generalized Fourier Series from Its Sum on a Compact Zero-Dimensional Group in the Non-Abelian Case

Analogue of Hardy-Littlewood Test for Fourier Series over Vilenkin System in the Case of Unbounded pk

Series with respect to a system of characters of a zero-dimensional compact commutative group are considered. A generalization of an analogue of Hardy-Littlewood test and its consequences, which were earlier obtained in the case of bounded sequence {pk} defining the system, are proved.

Some Convergence Tests for Fourier Series with Respect to Vilenkin System in the Case of Unbounded pk

Series with respect to a system of characters of a zero-dimensional compact Abelian group are considered. A generalization of an analogue of Dini test and its corollary obtained earlier for systems determined by bounded sequences {pk} are proved.

The dynamical zeta function for commuting automorphisms of zero-dimensional groups

For a $\mathbb{Z}^{d}$-action $\unicode[STIX]{x1D6FC}$ by commuting homeomorphisms of a compact metric space, Lind introduced a dynamical zeta function that generalizes the dynamical zeta function of a single transformation. In this article, we investigate this function when $\unicode[STIX]{x1D6FC}$ is generated by continuous automorphisms of a compact abelian zero-dimensional group. We address Lind’s conjecture concerning the existence of a natural boundary for the zeta function and prove this for two significant classes of actions, including both zero entropy and positive entropy examples. The finer structure of the periodic point counting function is also examined and, in the zero entropy case, we show how this may be severely restricted for subgroups of prime index in $\mathbb{Z}^{d}$. We also consider a related open problem concerning the appearance of a natural boundary for the dynamical zeta function of a single automorphism, giving further weight to the Pólya–Carlson dichotomy proposed by Bell and the authors.

On U- and M-sets for series with respect to characters of compact zero-dimensional groups

Generalizing our previous results on Walsh system, we consider the system of characters of zero-dimensional compact abelian group with the second axiom of countability and give for the sets of uniqueness with respect to this system a solution of category problem. We also obtain a generalization of a theorem on existence for this system a perfect M0-set whose Hausdorff h-measure equals zero.

Right ideal decompositions of G⁎

Let G be an infinite discrete group, let βG be the Stone–Čech compactification of G, and let G⁎=βG∖G. We show that if G can be embedded algebraically in a compact zero dimensional second countable group (in particular, if G=Z), then there are a decomposition D of G⁎ into right ideals of βG and a closed subsemigroup T of G⁎ containing all the idempotents such that DT={R∩T:R∈D} is a decomposition of T into closed right ideals and T/DT is homeomorphic to ω⁎.

Association schemes on general measure spaces and zero-dimensional Abelian groups

Association schemes form one of the main objects of algebraic combinatorics, classically defined on finite sets. At the same time, direct extensions of this concept to infinite sets encounter some problems even in the case of countable sets, for instance, countable discrete Abelian groups. In an attempt to resolve these difficulties, we define association schemes on arbitrary, possibly uncountable sets with a measure. We study operator realizations of the adjacency algebras of schemes and derive simple properties of these algebras. However, constructing a complete theory in the general case faces a set of obstacles related to the properties of the adjacency algebras and associated projection operators. To develop a theory of association schemes, we focus on schemes on topological Abelian groups where we can employ duality theory and the machinery of harmonic analysis. Using the language of spectrally dual partitions, we prove that such groups support the construction of general Abelian (translation) schemes and establish properties of their spectral parameters (eigenvalues).Addressing the existence question of spectrally dual partitions, we show that they arise naturally on topological zero-dimensional Abelian groups, for instance, Cantor-type groups or the groups of p-adic numbers. This enables us to construct large classes of examples of dual pairs of association schemes on zero-dimensional groups with respect to their Haar measure, and to compute their eigenvalues and intersection numbers (structural constants). We also derive properties of infinite metric schemes, connecting them with the properties of the non-Archimedean metric on the group.Next we focus on the connection between schemes on zero-dimensional groups and harmonic analysis. We show that the eigenvalues have a natural interpretation in terms of Littlewood–Paley wavelet bases, and in the (equivalent) language of martingale theory. For a class of nonmetric schemes constructed in the paper, the eigenvalues coincide with values of orthogonal function systems on zero-dimensional groups. We observe that these functions, which we call Haar-like bases, have the properties of wavelet bases on the group, including in some special cases the self-similarity property. This establishes a seemingly new link between algebraic combinatorics and (non-Archimedean) harmonic analysis.We conclude the paper by studying some analogs of problems of classical coding theory related to the theory of association schemes.

On M0-sets for series with respect to characters of compact zero-dimensional

Abstract Generalizing our previous result on Walsh system, we consider the system of characters of zero-dimensional compact abelian group. For this system, we give a construction of a perfect M0-set set whose Hausdorff h-measure equals zero, where h is any non-decreasing right-continuous function taking the value zero at the origin.

Riesz multiresolution analysis on zero-dimensional groups

We construct wavelet Riesz bases with non- compact support on zero- dimensional groups. For Vilenkin groups, we obtain an algorithm for constructing refinable functions in terms of trees.

On orthogonal p-adic wavelet bases

A variety of different orthogonal wavelet bases has been found for L2(R) for the last three decades. Such bases are not only interesting by themselves, a number of theoretical and applied problems were solved using them. It appeared that similar non-trivial wavelet constructions also exist for functions defined on some other algebraic structures, such as local fields of positive characteristic including the Vilenkin/Cantor groups. We note that most of the bases involved consist of Bruhat–Schwartz functions, i.e. compactly supported and band-limited ones. There were several attempts to develop wavelet theory in a more general setting, e.g., for zero-dimensional groups. Analyzing these papers, one can see that actually nothing except for the Haar basis and its trivial modifications was constructed. In the present paper we give an explanation for these failures. Namely, we describe the situation for the field Qp of p-adic numbers and prove that any orthogonal wavelet basis for L2(Qp) that consists of band-limited (periodic) functions, is a modification of the Haar basis.

Об операторе дифференцирования на компактных нуль-мерных группах

Saratov State University, 83, Astrakhanskaya str., Saratov, 410012, Russia, KrussUS@gmail.comWe define strong derivative on zero-dimensional compact group and find conditions under which the differential operator does notdepend from an orthonormal system that defines this derivative. For multidimensional case we find conditions under which thedifferential operator does not depend from method of conversion multidimensional group in one-dimensional group. We obtain aclear view of annihilators in a multidimensional compact zero-dimensional group.Key words: zero-dimensional groups, pseudo-differential operator, strong P-adic derivative, strong P-adic integral.

О множествах единственности кратных рядов по системе характеров нуль-мерной группы в смысле сходимости по кубам

О множествах единственности кратных рядов по системе характеров нуль-мерной группы в смысле сходимости по кубам

On the best approximations of functions defined on zero-dimensional groups

We present a survey of results obtained by the author, his disciples, and other mathematicians and related to the problem of finding the best approximations of functions in the investigation of properties of spaces of functions defined on zero-dimensional compact commutative groups.

Why are Haar bases in various structures the same?

The Haar system [1] is a model example of a wavelet basis in L2([0, 1]). The active study of wavelets began at the end of the twentieth century in connection with the development of multiresolution analysis. Soon interest in wavelet bases defined on various structures arose. In [2] and [3], a theory of multiresolution analysis on the Cantor group was developed, and various wavelet bases substantially different from Haar bases were found. The group of 2-adic numbers is largely similar to the Cantor group (in that they have the same canonical representation of elements and the same metric); the difference is only in the group operation “+,” but this results in dramatically different wavelet theories and multiresolution analyses. In particular, it is shown in [4] that, in contrast to the case of the Cantor group, there does not exist a p-adic multiresolution analysis with orthogonal scaling function other than Haar multiresolution analysis. At the same time, the p-adic Haar basis (first written out in [5]) has exactly the same form as the Haar basis on the Cantor group for p = 2 or on the Vilenkin group for p > 2. Haar bases on the interval [0, 1] with dilation factor pj depending on the level number j were studied in [6]. Haar bases on the rings of integer p-adic numbers and on the projective limits of finite cyclic groups in the oneand many-dimensional cases were constructed in [7]–[10]. The resemblance between the Haar basis on the zero-dimensional group and the Haar basis [6] on [0, 1] was indicated in [8, Remark 2.4]. In the present paper, we give a simple explanation of this resemblance by showing that all the above-mentioned Haar bases can be defined by the same formulas. We describe a sufficiently general scheme for constructing Haar bases which does not even rely on the presence of the group operation “+.” (Although the use of the group operation makes the notation of the Haar functions convenient, it can be completely avoided.) One only needs a “not too bad” topology on a measure space and a suitable sequence of partitions of the space. The construction of the Haar system and the proof of its basis property are exactly the same as for the classical Haar basis. Let μ be a measure on Ω, let Σ be the σ-algebra of measurable sets, and let p = {pj}j=−∞ be a sequence of integers pj > 1. Assume that there exist set collections {Ωjn}, n ∈ Z+, pairwise disjoint for each j ∈ Z and such that μΩ0n = 1 for each n ∈ Z+, Ω = ⋃ nΩjn for each j ∈ Z, and every Ωj−1,n is partitioned into pj subsets Ωj,nk all of which have the same measure, nk = nk(n, j) ∈ Z+. The family of pairs (j, n), numbering the sets Ωjn will be denoted by I, the set {Ωjn}(j,n)∈I will be called an (H,p)-partition, and its elements will be called cells. Let φjn be the characteristic function of the set Ωjn multiplied by a factor normalizing in L2(Ω). Set

Multiresolution analysis on product of zero-dimensional Abelian groups

For a product of locally compact zero-dimensional groups ( G , ∔ ) , we describe dilation operators and build the corresponding multiresolution analysis. We prove also that any nonsingular matrix A d × d over the field Z p defines a dilation operator in G d .

Multiresolution analysis on zero-dimensional Abelian groups and wavelets bases

For a locally compact zero-dimensional group , we build a multiresolution analysis and put forward an algorithm for constructing orthogonal wavelet bases. A special case is indicated when a wavelet basis is generated from a single function through contractions, translations and exponentiations.Bibliography: 19 titles.

Inversion formulae for the integral transform on a locally compact zero-dimensional group

Abstract Generalized inversion formulae for multiplicative integral transform with a kernel defined by characters of a locally compact zero-dimensional abelian group are obtained using a Kurzweil-Henstock type integral.

О рядах Хаара на компактной нуль-мерной группе

О рядах Хаара на компактной нуль-мерной группе