A matriceal analogue of Fejer's theory

J. Arazy [1] pointed out that there is a similarity between functions defined on the torus and infinite matrices. In this paper we discuss and develop in the framework of matrices Fejer's theory for Fourier series.     

Wiener's lemma for infinite matrices

The classical Wiener lemma and its various generalizations are important and have numerous applications in numerical analysis, wavelet theory, frame theory, and sampling theory. There are many different equivalent formulations for the classical Wiener lemma, with an equivalent formulation suitable for our generalization involving commutative algebra of infinite matrices . In the study of spline approximation, (diffusion) wavelets and affine frames, Gabor frames on non-uniform grid, and non-uniform sampling and reconstruction, the associated algebras of infinite matrices are extremely non-commutative, but we expect those non-commutative algebras to have a similar property to Wiener's lemma for the commutative algebra . In this paper, we consider two non-commutative algebras of infinite matrices, the Schur class and the Sjöstrand class, and establish Wiener's lemmas for those matrix algebras.

     

Some infinite classes of Hadamard matrices

Some results of Williamson [Duke Math. J., 11 1944, Bull. Amer. Math. Soc., 53 1947] and Wallis (J. Combinatorial Theory, 6 1969] are considerably improved to establish that in each case referred to, the same stated condition or conditions, which according to either of the authors give rise to one Hadamard matrix, actually imply the existence of an infinite series of Hadamard matrices. Also proved is the existence of some infinite series of Williamson's matrices, which coupled with the interesting findings of Turyn [J. Combinatorial Theory Ser. A, 16 1974] establish the existence of infinitely many more series of Hadamard matrices than those known so far.     

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.     

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.     

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.     

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.     

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.     

A new characterization of Bergman–Schatten spaces and a duality result

Let denote the space of all upper triangular matrices A such that lim_r→1⁻(1−r²)(A*C(r))^′_B(ℓ²)=0. We also denote by the closed Banach subspace of consisting of all upper triangular matrices whose diagonals are compact operators. In this paper we give a duality result between and the Bergman–Schatten spaces . We also give a characterization of the more general Bergman–Schatten spaces , 1p<∞, in terms of Taylor coefficients, which is similar to that of M. Mateljevic and M. Pavlovic [M. Mateljevic, M. Pavlovic, L^p-behaviour of the integral means of analytic functions, Studia Math. 77 (1984) 219–237] for classical Bergman spaces.     

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}．     

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     

On the distribution singular values of Toeplitz matrices

We prove a second order formula concerning distribution of singular values of Toeplitz matrices in some cases when conditions of the H. Widom Theorem are not satisfied.

     

A new infinite family of minimally nonideal matrices

Minimally nonideal matrices are a key to understanding when the set covering problem can be solved using linear programming. The complete classification of minimally nonideal matrices is an open problem. One of the most important results on these matrices comes from a theorem of Lehman, which gives a property of the core of a minimally nonideal matrix. Cornuéjols and Novick gave a conjecture on the possible cores of minimally nonideal matrices. This paper disproves their conjecture by constructing a new infinite family of square minimally nonideal matrices. In particular, we show that there exists a minimally nonideal matrix with r ones in each row and column for any r?3.