Nonstandard analysis and locally compact Abelian groups

Using nonstandard analysis, a new method is introduced for locally compact Abelian topological groups. Hyperfiniteness plays an important role. All LCA groups can be obtained this way. This yields approximation results for the (generalized) Fourier transform.     

Limits in compact Abelian groups

For X a compact Abelian group and B an infinite subset of its dual , let CB be the set of all x∈X such that converges to 1. If F is a free filter on , let . The sets CB and DF are subgroups of X. CB always has Haar measure 0, while the measure of DF depends on F. We show that there is a filter F such that DF has measure 0 but is not contained in any CB. This generalizes previous results for the special case where X is the circle group.     

Nonstandard representations of locally compact groups

In the note, it is proved that, under natural conditions, any infinite-dimensional unitary representation T of a direct product of groups G = K × N, where K is a compact group and N is a locally compact Abelian group, is imaged by a representation of the nonstandard analog \(\tilde G\) of the group G in the group of nonstandard matrices of a fixed nonstandard size.     

Convolution functions on compact Abelian groups

ПустьG — бесконечная компактная абелева г руппа с группой характеров?, и дляr>0A _r (G) обозначает множест во всехf∈L ₁ (G), преобразование Фурь е которыхf принадлеж итl _r (?). Пусть, далее, дляr>0 иs>0A(r, s)(G) обозначает множество всех такихf∈L ₁ (G), чтоf принадлежит про странству ЛоренцаL(r,s)(?). Теорема 1. Пусть 1<р≦2, 1<q≦2и 1/r=1/р+ 1/q?1. ТогдаL _p (G)*L _q (G)?A _r ,(G), 1/r+1/r′=1, причем равенство имеет место в том и тол ько том случае, когда p=q=2. Теорема 2. Пусть p, q, r удовл етворяют условиям те оремы 1 и 1/s=1/p+1/q. Тогда

1. существуютf∈L _p (G) и h∈L_q(G) т акие, чтоf*h ? A(β,γ)(G) ни для какихβ0;
2. если 0<s ₀ , то существую тf ∈L _p (G) и h ∈L _q ,(G) такие, что f*h∈A(r′, s ₀)(G).

Из теоремы 2 следует, чт о неравенство Юнга в определенном смысле неулучшаемо. Непосредственными с ледствиями теоремы 2 я вляются также один результат Р. Л. Липсмана и теорема У. Б. Тевари—А. К. Гупта.     

Lehmer's problem for compact Abelian groups

We formulate Lehmer's Problem concerning the Mahler measure of polynomials for general compact abelian groups, introducing a Lehmer constant for each such group. We show that all nontrivial connected compact groups have the same Lehmer constant and conjecture the value of the Lehmer constant for finite cyclic groups. We also show that if a group has infinitely many connected components, then its Lehmer constant vanishes.

     

Differences of functions and groups generators for compact Abelian groups

LetG be a Hausdorff compact Abelian group andC be the component of the identity element ofG. We consider a special class, ?(G), of functions inL ² (G) whose Fourier series satisfy certain convergence conditions (stronger than absolute convergence). We show thatG/C is topologically generated by not more thann elements if and only if, for each functionf in ?(G), there area ₁,...,a _n inG and functionf ₁,...f _n in ?(G) such that $$f = \sum\limits_{j = 1}^n {(f_j - \delta _{aj} * f_j ),}$$ where * is convolution defined in the usual sense, and δ _a denotes the Dirac measure ataεG.     

Limit theorems on locally compact Abelian groups

We prove limit theorems for row sums of a rowwise independent infinitesimal array of random variables with values in a locally compact Abelian group. First we give a proof of Gaiser's theorem [4, Satz 1.3.6], since it does not have an easy access and it is not complete. This theorem gives sufficient conditions for convergence of the row sums, but the limit measure cannot have a nondegenerate idempotent factor. Then we prove necessary and sufficient conditions for convergence of the row sums, where the limit measure can be also a nondegenerate Haar measure on a compact subgroup. Finally, we investigate special cases: the torus group, the group of p ‐adic integers and the p ‐adic solenoid. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extremal pseudocompact Abelian groups are compact metrizable

Every pseudocompact Abelian group of uncountable weight has both a proper dense pseudocompact subgroup and a strictly finer pseudocompact group topology.

     

Varieties generated by countably compact Abelian groups

We prove under the assumption of Martin's Axiom that every precompact Abelian group of size belongs to the smallest class of groups that contains all Abelian countably compact groups and is closed under direct products, taking closed subgroups and continuous isomorphic images.

     

Mixing rank-one actions of locally compact Abelian groups

Using techniques related to the (C,F)-actions we construct explicitly mixing rank-one (by cubes) actions T of G=Rd₁×Zd₂ for any pair of non-negative integers d₁, d₂. It is also shown that h(Tg)=0 for each g∈G.     

On homogeneous stochastic processes on compact Abelian groups

This article discusses the extension to general compact Abelian groups of some results previously established by R. Roy for the case of the circle and the sphere. Estimators of the covariance function and spectral parameters for a homogeneous stochastic process defined on a compact Abelian group are considered and their properties are derived.     

The Heyde theorem for locally compact Abelian groups

We prove a group analogue of the well-known Heyde theorem where a Gaussian measure is characterized by the symmetry of the conditional distribution of one linear form given another. Let X be a locally compact second countable Abelian group containing no subgroup topologically isomorphic to the circle group T, G be the subgroup of X generated by all elements of order 2, and Aut(X) be the set of all topological automorphisms of X. Let αj,βj∈Aut(X), j=1,2,…,n, n?2, such that for all i≠j. Let ξj be independent random variables with values in X and distributions μj with non-vanishing characteristic functions. If the conditional distribution of L₂=β₁ξ₁+?+βnξn given L₁=α₁ξ₁+?+αnξn is symmetric, then each μj=γj∗ρj, where γj are Gaussian measures, and ρj are distributions supported in G.     

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.