Wiener's lemma for twisted convolution and Gabor frames

We prove non-commutative versions of Wiener's Lemma on absolutely convergent Fourier series (a) for the case of twisted convolution and (b) for rotation algebras. As an application we solve some open problems about Gabor frames, among them the problem of Feichtinger and Janssen that is known in the literature as the ``irrational case'.

     

Johnson‐Lindenstrauss lemma for circulant matrices**

We prove a variant of a Johnson‐Lindenstrauss lemma for matrices with circulant structure. This approach allows to minimize the randomness used, is easy to implement and provides good running times. The price to be paid is the higher dimension of the target space k = O(ε^?2 log³ n) instead of the classical bound k = O(ε^?2 log n). © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Some special classes of infinite matrices

In this paper, we present a simple method to construct examples of infinite matrices belonging to different classes of matrix spaces. Moreover, we introduce a scale of matrix spaces which extends the well-known scale of classical Lebesgue spaces.     

Wiener＇s lemma：localization and various approaches

Matrices and integral operators with off-diagonal decay appear in numerous areas of mathematics including numerical analysis and harmonic analysis, and they also play impor-tant roles in engineering science including signal processing and communication engineering. Wiener’s lemma states that the localization of matrices and integral operators are preserved un-der inversion. In this introductory note, we re-examine several approaches to Wiener’s lemma for matrices. We also review briefly some recent advances on localization preservation of operations including nonlinear inversion, matrix factorization and optimization.     

An almost periodic noncommutative Wiener's Lemma

We develop a theory of almost periodic elements in Banach algebras and present an abstract version of a noncommutative Wiener's Lemma. The theory can be used, for example, to derive some of the recently obtained results in time-frequency analysis such as the spectral properties of the finite linear combinations of time-frequency shifts.     

Perron-Frobenius operators and representations of the Cuntz-Krieger algebras for infinite matrices

In this paper we extend the work of Kawamura, see [K. Kawamura, The Perron-Frobenius operators, invariant measures and representations of the Cuntz-Krieger algebras, J. Math. Phys. 46 (2005)], for Cuntz-Krieger algebras OA for infinite matrices A. We generalize the definition of branching systems, prove their existence for any given matrix A and show how they induce some very concrete representations of OA. We use these representations to describe the Perron-Frobenius operator, associated to a nonsingular transformation, as an infinite sum and under some hypothesis we find a matrix representation for the operator. We finish the paper with a few examples.     

Derivations of the Lie algebra of infinite strictly upper triangular matrices over a commutative ring

Let 𝒩(∞,R) be the Lie algebra of infinite strictly upper triangular matrices over a commutative ring R. We show that every derivation of 𝒩(∞,R) is a sum of diagonal and inner derivations.     

Population dynamics with infinite Leslie matrices: finite time properties

Infinite Leslie matrices, introduced by Demetrius 40 years ago, are mathematical models of age-structured populations defined by a countable infinite number of age classes. This article is concerned with determining solutions of the discrete dynamical system in finite time. We address this problem by appealing to the concept of kneading matrices and kneading determinants. Our analysis is applicable not only to populations models, but also to models of self-reproducing machines and self-reproducing computer programs. The dynamics of these systems can also be described in terms of infinite Leslie matrices.     

Three linear preservers on infinite triangular matrices

We consider three linear preserver problems on the algebra of infinite triangular matrices over fields. We characterize the maps preserving invertible and noninvertible matrices, the surjective maps preserving inverses and the surjective maps preserving rank permutability.     

Left inverses of matrices with polynomial decay

It is known that the algebra of Schur operators on ?² (namely operators bounded on both ?¹ and ?^∞) is not inverse-closed. When ?²=?²(X) where X is a metric space, one can consider elements of the Schur algebra with certain decay at infinity. For instance if X has the doubling property, then Q. Sun has proved that the weighted Schur algebra Aω(X) for a strictly polynomial weight ω is inverse-closed. In this paper, we prove a sharp result on left-invertibility of the these operators. Namely, if an operator A∈Aω(X) satisfies ‖Afp_‖?‖fp_‖, for some 1?p?∞, then it admits a left-inverse in Aω(X). The main difficulty here is to obtain the above inequality in ?². The author was both motivated and inspired by a previous work of Aldroubi, Baskarov and Krishtal (2008) [1], where similar results were obtained through different methods for X=Zd, under additional conditions on the decay.     

对称无限布尔方阵的本原指数的一个计算公式

引入了本原无限布尔方阵的概念,给出了对称无限布尔方阵为本原阵的一个充分必要条件,最后给出了对称本原无限布尔方阵的本原指数的一个计算公式.     

围长为2的本原无限布尔方阵类的本原指数集

研究了围长为2的无限布尔方阵的本原性,通过无限有向图D（A）的直径给出了这类矩阵的本原指数的上确界,最后证明了直径小于等于d且围长为2的本原无限布尔方阵所构成的矩阵类的本原指数集为Ed^0={2,3,…,3d}．     

A variant of the Johnson-Lindenstrauss lemma for circulant matrices

We continue our study of the Johnson-Lindenstrauss lemma and its connection to circulant matrices started in Hinrichs and Vybíral (in press) [7]. We reduce the bound on k from k=Ω(ε−2log³n) proven there to k=Ω(ε−2log²n). Our technique differs essentially from the one used in Hinrichs and Vybíral (in press) [7]. We employ the discrete Fourier transform and singular value decomposition to deal with the dependency caused by the circulant structure.     

Diagonalization of row-column-finite infinite matrices

A complete solution to the diagonalization problem (under equivalence) for row-column-finite infinite matrices over a general field is given Project supported by the National Natural Science Foundation of China     

Symmetry and inverse-closedness of matrix algebras and functional calculus for infinite matrices

We investigate the symbolic calculus for a large class of matrix algebras that are defined by the off-diagonal decay of infinite matrices. Applications are given to the symmetry of some highly non-commutative Banach algebras, to the analysis of twisted convolution, and to the theory of localized frames.