Spectral features and asymptotic properties for α-circulants and α-Toeplitz sequences: theoretical results and examples

For a given nonnegative integer α, a matrix An of size n is called α-Toeplitz if its entries obey the rule An = [ar−αs] n−1 r,s=0. Analogously, a matrix An again of size n is called α-circulant if An = [ a(r−αs) modn ]n−1 r,s=0 . Such kind of matrices arises 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 α-circulants and we provide an asymptotic analysis of the distribution results for the singular values of α-Toeplitz sequences in the case where {ak} can be interpreted as the sequence of Fourier coefficients of an integrable function f over the domain (−π, π). Some generalizations to the block, multilevel case, amounting to choose f multivariate and matrix valued, are briefly considered.