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

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

2009-06-11

arxiv.org/abs/0906.2104v1

Mathematics

Numerical Analysis

38 pages, submitted to SIMAX

Scientific paper

For a given nonnegative integer alpha, a matrix A_{n} of size n is called alpha-Toeplitz if its entries obey the rule A_{n}=[a_{r-alpha*s}]_{r,s=0}^{n-1}. Analogously, a matrix A_{n} again of size n is called alpha-circulant if A_{n}= [a_{(r-alpha*s)mod n}]_{r,s=0}^{n-1}. 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 alpha-circulants and we provide an asymptotic analysis of the distribution results for the singular values of alpha-Toeplitz sequences in the case where {a_{k}} can be interpreted as the sequence of Fourier coeffcients of an integrable function f over the domain (-pi;pi). Some generalizations to the block, multilevel case, amounting to choose f multivariate and matrix valued, are briefly considered.

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