Band 5, Heft 4

This paper is concerned with a new type of matrix. This class of matrix is a generalization of the r -circulant matrix, and we call it ( r , s )-pair circulant matrix. Some properties such as sum, difference, product, inverse and adjoint of ( r , s )-pair circulant matrices are investigated. Moreover, we give some necessary and sufficient conditions for a matrix to be an ( r , s )-pair circulant matrix. Finally, an algorithm for computing the inverse or generalized inverse of an ( r , s )-pair circulant matrix is presented, some numerical examples are also given.

In this paper, we will prove the superstability of the functional equation ∑ϕ∈Φ∫Kf(xkϕ(y)k-1)dwK(k)=|Φ|f(x)g(y),x,y∈G,$ \sum _{\varphi \in \Phi }\int _{K} f(xk\varphi (y)k^{-1})dw_{K}(k) =\vert \Phi \vert f(x)g(y), \quad x,y\in G, $ which is bounded by the real functions φ(x)${\phi (x)}$, ψ(y)${\psi (y)}$ or min(φ(x),ψ(y))${\min (\phi (x),\psi (y))}$, where f , g are unknown continuous complex-valued functions. Here G is any topological group, K is a compact subgroup of G , ωK${\omega _{K}}$ is the normalized Haar measure of K and Φ is a finite group of K -invariant morphisms of G . Consequently, we will investigate the superstability of the generalized Pexider type trigonometric functional equation ∑ϕ∈Φ∫Kf(xkϕ(y)k-1)dwK(k)=|Φ|g(x)h(y),x,y∈G,$ \sum _{\varphi \in \Phi }\int _{K}f(xk\varphi (y)k^{-1})dw_{K}(k) =\vert \Phi \vert g(x)h(y), \quad x,y\in G, $ and we give some applications.

Let R be a ring on an Abelian group A . The symbol R n denotes the set of all finite sums of products of n elements of R . The smallest positive integer n such that R n = 0 is called the index of nilpotency of R and is denoted by I ( R ) = n . In this paper, we study some nilpotent rings. In particular, we determine the index of nilpotency of some torsion-free rings of finite rank.

We investigate the connection between compactness of radial operators on the generalized Fock spaces and the behavior at infinity of the corresponding Berezin transform and give a condition under which compactness of an operator is equivalent to the vanishing of the Berezin transform at infinity. Our result is based on a certain o-Tauberian theorem. We also show that the eigenvalues of radial operators depend only on the length of the multi-index.

The goal is to construct random frames and study properties of such frames. Starting with the construction of unimodular random sequences whose expected autocorrelations can be made arbitrarily low outside the origin, these random sequences are used to construct frames for ℂ d . Using recent theory of non-asymptotic analysis of random matrices, the eigenvalue distribution of the corresponding frame operator is studied.