Zero-dimensional Abelian Group Research Articles

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 spectral synthesis on zero-dimensional Abelian groups

Let be a zero-dimensional locally compact Abelian group all of whose elements are compact, and let be the space of all complex-valued continuous functions on . A closed linear subspace is said to be an invariant subspace if it is invariant with respect to the translations , . In the paper, it is proved that any invariant subspace admits spectral synthesis, that is, coincides with the closed linear span of the characters of belonging to .Bibliography: 25 titles.

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 .

Haar system on a product of zero-dimensional compact groups

Abstract In this work, we study the problem of constructing Haar bases on a product of arbitrary compact zero-dimensional Abelian groups. A general scheme for the construction of Haar functions is given for arbitrary dimension. For dimension d=2, we describe all Haar functions.

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.

Denjoy and P-path integrals on compact groups in an inversion formula for multiplicative transforms

Abstract Denjoy and P-path Kurzweil-Henstock type integrals are defined on compact subsets of some locally compact zero-dimensional abelian groups. Those integrals are applied to obtain an inversion formula for the multiplicative integral transform.

О рядах Хаара на компактной нуль-мерной группе

О рядах Хаара на компактной нуль-мерной группе

Kurzweil-Henstock type integral on zero-dimensional group and some of its applications

A Kurzweil-Henstock type integral on a zero-dimensional abelian group is used to recover by generalized Fourier formulas the coefficients of the series with respect to the characters of such groups, in the compact case, and to obtain an inversion formula for multiplicative integral transforms, in the locally compact case.

Perron type integral on compact zero-dimensional Abelian groups

Perron and Henstock type integrals defined directly on a compact zero-dimensional Abelian group are studied. It is proved that the considered Perron type integral defined by continuous majorants and minorants is equivalent to the integral defined in the same way, but without assumption on continuity of majorants and minorants.

Henstock type integral in harmonic analysis on zero-dimensional groups

A Henstock type integral is defined on compact subsets of a locally compact zero-dimensional abelian group. This integral is applied to obtain an inversion formula for the multiplicative integral transform.

Isomorphism and Measure Rigidity for Algebraic Actions on Zero-Dimensional Groups

We consider mixing ℤd-actions on compact zero-dimensional abelian groups by automorphisms. Rigidity of invariant measures does not hold for such actions in general; we present conditions which force an invariant measure to be Haar measure on an affine subset. This is applied to isomorphism rigidity for such actions. We develop a theory of halfspace entropies which plays a similar role in the proof to that played by invariant foliations in the proof of rigidity for smooth actions.

Isomorphism rigidity of irreducible algebraic ℤd-actions

An irreducible algebraic ℤ d -actionα on a compact abelian group X is a ℤ d -action by automorphisms of X such that every closed, α-invariant subgroup Y⊊X is finite. We prove the following result: if d≥2, then every measurable conjugacy between irreducible and mixing algebraic ℤ d -actions on compact zero-dimensional abelian groups is affine. For irreducible, expansive and mixing algebraic ℤ d -actions on compact connected abelian groups the analogous statement follows essentially from a result by Katok and Spatzier on invariant measures of such actions (cf. [4] and [3]). By combining these two theorems one obtains isomorphism rigidity of all irreducible, expansive and mixing algebraic ℤ d -actions with d≥2.

Instances of cohomological triviality and rigidity

AbstractWe consider a compact metric zero-dimensional abelian group X and mixing ℤ2 (or ℤ2+) actions by automorphisms (or endomorphisms). We provide various instances of cohomological triviality and rigidity for cocycles taking values in finite groups.

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.