Spectral Features and Asymptotic Properties for g-Circulants and g-Toeplitz Sequences

Abstract

For a given nonnegative integer g, a matrix $A_n$ of size n is called g-Toeplitz if its entries obey the rule $A_n=[a_{r-gs}]_{r,s=0}^{n-1}$. Analogously, a matrix $A_n$ again of size n is called g-circulant if $A_n=\bigl[a_{(r-g s) \mathrm{mod}\,n}\bigr]_{r,s=0}^{n-1}$. Such matrices arise in wavelet analysis, subdivision algorithms, and more generally when dealing with multigrid/multilevel methods for structured matrices and approximations of boundary value problems. In this paper we study the singular values of g-circulants and provide an asymptotic analysis of the distribution results for the singular values of g-Toeplitz sequences in the case where $\{a_k\}$ can be interpreted as the sequence of Fourier coefficients of an integrable function f over the domain $(-\pi,\pi)$. Generalizations to the block and multilevel case are also considered.