Asymptotic eigenvalue distribution of block-Toeplitz matrices

Abstract

The need for approximating block-Toeplitz with Toeplitz block matrices by means of block-circulant with circulant block matrices with the objective of transforming an inherently ill-posed image deconvolution problem to a well-posed one motivated a surge of papers on the analysis of the quality as well as the speed of convergence of the algorithms that produce such approximants to serve as preconditioners for conjugate-gradient methods. This correspondence contributes to that surge by giving a simple proof of the fact that the sequence of eigenvalues of the Hermitian block Toeplitz with Toeplitz-block matrices are asymptotically equidistributed. To do this, Weyl's (1916) results on the distribution properties of multidimensional sequences are exploited. Inferences to related results are made.