Symbol approach in a signal-restoration problem involving block Toeplitz matrices

Abstract

We consider a special type of signal restoration problem where some of the sampling data are not available. The formulation related to samples of the function and its derivative leads to a possibly large linear system associated to a nonsymmetric block Toeplitz matrix which can be equipped with a 2×2 matrix-valued symbol. The aim of the paper is to study the eigenvalues of the matrix. We first identify in detail the symbol and its analytical features. Then, by using recent results on the eigenvalue distribution of block Toeplitz matrix-sequences, we formally describe the cluster sets and the asymptotic spectral distribution of the matrix-sequences related to our problem. The localization areas, the extremal behavior, and the conditioning are only observed numerically, but their behavior is strongly related to the analytical properties of the symbol, even though a rigorous proof is still missing in the block case.