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.     

离散交换群上的Wiener-Hopf算子(英)

用groupoid方法研究了离散交换群上的Wiener－Hopf算子．     

Fundamental Sets of Functions on Locally Compact Abelian Groups

Let G be a locally compact Abelian group, and let X be a compact set of G. Given a positive definite function ?: G × G → ? whose real part is continuous at neutral element of G, we research a necessary and sufficient setting for the linear span of the set {x ∈ X → ?(x ? y): y ∈ X} to be dense in C(X) in the topology of uniform convergence. The context treated that is abstract encompasses classical cases of the literature, while other examples are entirely new.     

On the Skitovich-Darmois Theorem on Abelian Groups

We prove theorems that generalize the Skitovich-Darmois theorem to the case where independent random variables ξ_j, j = 1, 2, ..., n, n ≥ 2, take values in a locally compact Abelian group and the coefficients α_j and β_j of the linear forms L ₁ = α₁ξ₁ + ... + α_nξ_n and L ₂ = β₁ξ₁ + ... + β_nξ_n are automorphisms of this group.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 10, pp. 1342 – 1356, October, 2004.     

Discrete Systems and their Characteristic Spectral Functions

We define first-order discrete systems in the matrix-valued case. They are characterized by sequences of pair of matrices, called admissible sequences. We present two important examples of such sequences, called Szeg? and Nehari sequences. We introduce the characteristic spectral functions associated to a first-order system. We define in particular the scattering function, the Weyl function and the reflection coefficient function and we study the relationships between these functions. Daniel Alpay wishes to thank the Earl Katz family for endowing the chair which supported his research.     

On Purity and Quasiequality of Abelian Groups

We study the Cohn purity in an abelian group regarded as a left module over its endomorphism ring. We prove that if a finite rank torsion-free abelian group G is quasiequal to a direct sum in which all summands are purely simple modules over their endomorphism rings then the module _E(G) G is purely semisimple. This theorem makes it possible to construct abelian groups of any finite rank which are purely semisimple over their endomorphism rings and it reduces the problem of endopure semisimplicity of abelian groups to the same problem in the class of strongly indecomposable abelian groups.     

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.     

On Zeta-Functions of Finite Abelian Groups

In the paper, the zeta-functions of finite Abelian groups of rank 2 and 3 are considered. One- and two-dimensional limit theorems for these zeta-functions on the complex plane and in the space of meromorphic functions are obtained.     

On Schurity of Finite Abelian Groups

A finite group G is called a Schur group, if any Schur ring over G is associated in a natural way with a subgroup of Sym(G) that contains all right translations. Recently, the authors have completely identified the cyclic Schur groups. In this article, it is shown that any abelian Schur group belongs to one of several explicitly given families only. In particular, any noncyclic abelian Schur group of odd order is isomorphic to ?₃ × ?_{3 ^k} or ?₃ × ?₃ × ? _p where k ≥ 1 and p is a prime. In addition, we prove that ?₂ × ?₂ × ? _p is a Schur group for every prime p.     

关于有限Abel p-群的自同构群

从有限Abel p-群P的型不变量出发,给出了其自同构群AutP的阶的计算公式,并利用|AutP|的计算公式得到了下面3个结果:1.由有限Abel p-群的型不变量的两种变换得到了其自同构群的阶的变化规律;2.用群的阶、秩、幂指数三个量界定了有限Abel p-群的自同构的阶;3.对部分Frattini子群为p阶群的有限p-群,确定了其自同构群的阶何时达到最小值和最大值.     

可除阿贝尔p-群的自同构群

主要探讨了秩大于或者等于p-1的可除阿贝尔p-群的p-自同构群,并且得到这些p-自同构如何作用在该可除阿贝尔p-群上.这些结论有助于进一步理解Cernikov p-群的结构.     

On Associative Ring Multiplication on Abelian Mixed Groups

An abelian group is called a mixed one if it is neither torsion nor torsion-free. It is to be proved that every mixed group can be provided with a nonzero associative ring structure. Our methods of proofs are straightforward and elementary.     

Recovery of Band-Limited Functions on Locally Compact Abelian Groups from Irregular Samples

Using the techniques of approximation and factorization of convolution operators we study the problem of irregular sampling of band-limited functions on a locally compact Abelian group G. The results of this paper relate to earlier work by Feichtinger and Gröchenig in a similar way as Kluvánek's work published in 1969 relates to the classical Shannon Sampling Theorem. Generally speaking we claim that reconstruction is possible as long as there is sufficient high sampling density. Moreover, the iterative reconstruction algorithms apply simultaneously to families of Banach spaces.     

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 orders Solely of Abelian Groups II

Let C'(x) denote the number of integers n x such that thereis no non-abelian group of order n, but there exists a non-cyclicgroup of order n. Here it is shown that C' (x) , x , where denotes Euler's constant.     

On the Heyde Theorem for Finite Abelian Groups

It is well-known Heyde's characterization theorem for the Gaussian distribution on the real line: if _j are independent random variables, _j , _j are nonzero constants such that _i ± _j ^–1 _j 0 for all i j and the conditional distribution of L ₂=_{1 1} + ··· + _{n n} given L ₁=_{1 1} + ··· + _{n n} is symmetric, then all random variables _j are Gaussian. We prove some analogs of this theorem, assuming that independent random variables take on values in a finite Abelian group X and the coefficients _j , _j are automorphisms of X.     

Decay of Concentration Functions for Adapted Probabilities on Discrete Groups

Given a probability measure, , on a locally compact group, necessary and sufficient support conditions which ensure that the concentration functions associated with converge to zero have previously been determined. In this note the rate of this convergence when is adapted on a discrete group G is shown to depend on the volume growth rate of N , the smallest normal subgroup a coset of which contains the support of .