Compact Operators on Homogeneous Banach Spaces of Distributions on Locally Compact Abelian Groups

In this paper, two theorems about the compactness of almost invariant operators on homogeneous Banach spaces of distributions (in the sense of Feichtinger [11]) defined on a locally compact abelian group are proved. Our theorems generalize the corresponding results of K. de Leeuw [3] and Tewari and Madan [18] for operators on homogeneous Banach spaces on the circle group and Segal algebras on a compact abelian groups, respectively.AMS Subject Classification 2000: 43A85, 47B07.     

Abstract Inverse Estimation with Application to Deconvolution on Locally Compact Abelian Groups

Recovery of the unknown parameter in an abstract inverse estimation model can be based on regularizing the inverse of the operator defining the model. Such regularized-inverse type estimators are constructed with the help of a version of the spectral theorem due to Halmos, after suitable preconditioning. A lower bound to the minimax risk is obtained exploiting the van Trees inequality. The proposed estimators are shown to be asymptotically optimal in the sense that their risk converges to zero, as the sample size tends to infinity, at the same rate as this lower bound. The general theory is applied to deconvolution on locally compact Abelian groups, including both indirect density and indirect regression function estimation.     

Finite-Dimensional Approximations of Operators in the Hilbert Spaces of Functions on Locally Compact Abelian Groups

A new approach to the approximation of operators in the Hilbert space of functions on a locally compact Abelian (LCA) group is developed. This approach is based on sampling the symbols of such operators. To choose the points for sampling, we use the approximations of LCA groups by finite groups, which were introduced and investigated by Gordon. In the case of the group R ⁿ , the constructed approximations include the finite-dimensional approximations of the coordinate and linear momentum operators, suggested by Schwinger. The finite-dimensional approximations of the Schrödinger operator based on Schwinger's approximations were considered by Digernes, Varadarajan, and Varadhan in Rev. Math. Phys. 6 (4) (1994), 621–648 where the convergence of eigenvectors and eigenvalues of the approximating operators to those of the Schrödinger operator was proved in the case of a positive potential increasing at infinity. Here this result is extended to the case of Schrödinger-type operators in the Hilbert space of functions on LCA groups. We consider the approximations of p-adic Schrödinger operators as an example. For the investigation of the constructed approximations, the methods of nonstandard analysis are used.     

On the Approximation of Functions on Locally Compact Abelian Groups

Questions of approximative nature are considered for a space of functions L ^p(G, ), 1 p , defined on a locally compact abelian Hausdorff group G with Haar measure . The approximating subspaces which are analogs of the space of exponential type entire functions are introduced.     

Contractive Automorphisms of Locally Compact Groups and the Concentration Function Problem

Let G be a noncompact locally compact group. We show that a necessary and sufficient condition in order that G support an adapted probability measure whose concentration functions fail converge to zero is that G be the semidirect product , where is an automorphism of N contractive modulo a compact subgroup. Any adapted a probability measure whose concentration functions fail to converge to zero has the form =v×₁ where v is a probability measure on N. If G is unimodular then the concentration functions of an adapted probability measure fail to converge to zero if and only if is supported on a coset of a compact normal subgroup.     

Sixteen-Dimensional Locally Compact Translation Planes with Large Automorphism Groups having no Fixed Points

The 16-dimensional compact projective planes whose automorphism group contains a closed connected subgroup fixing a line, but no point and having dimension at least 35 are determined. It is shown that these planes all belong to three families of planes determined by H. Löwe and the author, and hence are explicitly known. A major stepping stone to this goal is a result by H. Salzmann according to which every such plane is a translation plane.     

Time Regularity for Random Walks on Locally Compact Groups

Let G be a compactly generated, locally compact group, and let T be the operator of convolution with a probability measure μ on G. Our main results give sufficient conditions on μ for the operator T to be analytic in L ^p (G), 1 < p < ∞, where analyticity means that one has an estimate of form for all n = 1, 2, ... in L ^p operator norm. Counterexamples show that analyticity may not hold if some of the conditions are not satisfied.     

Abel群的亚同态的刻画

In this paper,a characterization of metahomomorphisrns on Abeliangroups is given.     

Locally Asplund Preduals of Spaces of Holomorphic Functions

Defant [5] introduced the local Radon–Nikodym property for duals of locally convex spaces. This is a generalization of Asplund spaces as defined in Banach space theory. In this paper we generalise Dunford"s Theorem [7] to Banach spaces with Schauder decompositions and apply this result to spaces of holomorphic functions on balanced domains in a Banach space. This revised version was published online in June 2006 with corrections to the Cover Date.     

Rate of Decay of Concentration Functions on Discrete Groups

Given an irreducible probability measure on a non-compact locally compact group G, it is known that the concentration functions associated with converge to zero. In this note the rate of this convergence is presented in the case where G is a non-locally finite discrete group. In particular it is shown that if the volume growth V(m) of G satisfies V(m) cm ^D then for any compact set K we have sup _gG ⁽ⁿ⁾(Kg) Cn ^–D/2.     

Mappings on Spaces of Continuous Functions

Let X and Y be locally compact Hausdorff spaces and T : C₀(X) C₀(Y) a ring homomorphism. We completely characterize such homomorphisms and show that if T is R-linear, then T is either C-linear or C-antilinear. In any case T is continuous and there is a continuous map : Y X such that Tf = f o , f C₀(X) (if T is C-linear) or (if T is C-antilinear). Thus, extending a result of Mólnar, we also derive the general form of an isometry T.AMS Subject Classification (2000): primary 46J05, 46E25(deceased) Passed away on 24 May 1999.     

关于局部紧域的加法特征的一点注记

In this note, we give an elementary and constructive proof for that the additive character group of a locally compact field is isomorphic to itself as an additive topological group.     

Boundeness of a Class of Maximal Operators on H~p Spaces on Compact Lie Groups

In this paper we discuss the weak type(H^p,L^p) boundedness of a class of maximal operators T*^ψ and themaximal strong mean boundedness of a family of the operators ｛T^ψ｝ on the atomic H^p spaces on compact Lic groups.Also,we obtain the correspoding convergent results.     

抽象度量空间上的一致连续函数空间

本文主要讨论抽象度量空间上的一致连续函数空间的Banach空间结构,代数结构和格结构．     

The Influence of Some Quantitative Properties of Abelian Subgroups on Solvability of Finite Groups

As an important application of Thompson's theorem [9 Robinson , D. J. S. ( 1996 ). A Course in the Theory of Groups. , 2nd ed. New York : Springer-Verlag .[Crossref] , [Google Scholar], Theorem 10.4.2], a finite group is solvable if it has an abelian maximal subgroup. In this article, we mainly investigate the influence of some quantitative properties of abelian subgroups on solvability of finite groups. Some new results are obtained.     

Noninjectivity of the Predual Bimodule of the Measure Algebra for Infinite Discrete Groups

In this paper, it is proved that the predual bimodule of the measure algebra of an infinite discrete group is not injective despite the fact that the measure algebra of an amenable group is amenable in the sense of Connes. Thus the well-known result of Khelemskii (claiming that, for a von Neumann algebra, Connes-amenability is equivalent to the condition that the predual bimodule is injective) cannot be extended to measure algebras. Moreover, for a discrete amenable group, we give a simple formula for a normal virtual diagonal of the measure algebra. It is shown that a certain canonical bimodule over the measure algebra is not normal.     

Weak Mixing of Random Walks on Groups

Let G be a locally compact -compact group with right Haar measure m and a regular probability measure on G. We say that is weakly mixing if for all gL (G) and all fL ¹(G) with fdm=0 we have n ^{–1 n} _k=1| ^k *f,g|0. We show that is weakly mixing if and only if is ergodic and strictly aperiodic. To prove this we use and prove some results about unimodular eigenvalues for general Markov operators.     

Asymptotic Expansions of Hermite Functions on Compact Lie Groups

Let G be a compact, connected Lie group endowed with a bi-invariant Riemannian metric. Let _t be the heat kernel on G; that is, _t is the fundamental solution to the heat equation on the group determined by the Laplace–Beltrami operator. Recent work of Gross (1993) and Hijab (1994) has led to the study of a new family of functions on G. These functions, obtained from _t and its derivatives, are the compact group analogs of the classical Hermite polynomials on . Previous work of this author has established that these Hermite functions approach the classical Hermite polynomials on in the limit of small t, where the Hermite functions are viewed as functions on via composition with the exponential map. The present work extends these results by showing that these Hermite functions can be expanded in an asymptotic series in powers of . For symmetrized derivatives, it is shown that the terms with fractional powers of t vanish. Additionally, the asymptotic series for Hermite functions associated to powers of the Laplacian are computed explicitly. Remarkably, these asymptotic series terminate, yielding a polynomial in t.