Chang-Pao Chen

Operator norms and lower bounds of generalized Hausdorff matrices

Let A = (a n,k ) n,k≥0 be a non-negative matrix. Denote by L p,q (A) the supremum of those L satisfying the following inequality: The purpose of this article is to establish a Bennett-type formula for and a Hardy-type formula for and , where is a generalized Hausdorff matrix and 0 < p ≤ 1. Similar results are also established for and for other ranges of p and q. Our results extend [Chen and Wang, Lower bounds of Copson type for Hausdorff matrices, Linear Algebra Appl. 422 (2007), pp. 208–217] and [Chen and Wang, Lower bounds of Copson type for Hausdorff matrices: II, Linear Algebra Appl. 422 (2007) pp. 563–573] from to with α ≥ 0 and completely solve the value problem of , , and for α ∈ ℕ ∪ {0}.

The best constants for multidimensional modular inequalities over spherical cones

In this paper, we establish the Muckenhoupt-type estimation for the best constant associated with the following multidimensional modular inequality over a spherical cone:where and . Similar results are also derived for the complementary integral operator. Our results provide the -dimensional modular forms of the works of Andersen and Heinig. As consequences of our results, we give the -dimensional weighted extensions of Levinson modular inequality, extensions of Stepanov and Heinig results, generalizations of the Hardy–Knopp-type inequalities, and those for the Riemann–Liouville operator and the Weyl fractional operator. We also point out that our estimates are better than those given in the works of Drabek, Heinig, Kunfer, and Sinnamon.

The weighted norm inequalities of integral operators with special types of kernels

Abstract In this paper, the weighted norm inequalities of the following special type of integral operators are investigated: where or . In particular, those established by E. T. Copson, Hardy-Littlewood-Pólya, S. Lai, N. Levinson, E. R. Love, B. G. Pachpatte and Yang-Hwang will be derived.

Matrices Whose Norms Are Determined by Their Actions on Decreasing Sequences

Let A = (a j,k) j,k�1 be a non-negative matrix. In this paper, we characterize those A for which kAkE,F are determined by their actions on decreasing sequences, where E and F are suitable normed Riesz spaces of sequences. In particular, our results can apply to the following spaces: lp, d(w, p), and lp(w). The results established here generalizeones given by Bennett; Chen, Luor, and Ou; Jameson; and Jameson and Lashkaripour.

Lower bounds of Copson type for weighted mean matrices and Nörlund matrices

Let 1 ≤ p ≤ ∞, 0 < q ≤ p, and A = (a n,k ) n,k≥0 ≥ 0. Denote by L p,q (A) the supremum of those L satisfying the following inequality: whenever and X ≥ 0. In this article, the value distribution of L p,q (A) is determined for weighted mean matrices, Nörlund matrices and their transposes. We express the exact value of L p,q (A) in terms of the associated weight sequence. For Nörlund matrices and some kinds of transposes, this reduces to a quotient of the norms of such a weight sequence. Our results generalize the work of Bennett.

An acceleration method for computing the generalized eigenvalue problem on a parallel computer

Abstract We give a cubic correction step for improving the current eigenvalue algorithms for computing the generalized Schur decomposition of a regular pencil λB − A using a Jacobi-like method. The correction method can be used to speed up the convergence at the end of the Jacobi-like process when the strictly lower triangular elements of the matrix pair ( A , B ) have become sufficiently small; it can be implemented in parallel on an n × n square array of mesh-connected processors in O ( n ) computational time. A quantitative analysis of the convergence and a comparison of the complexity of one Jacobi sweep versus one correction step are presented.