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For a given nonnegative integer alpha, a matrix A_{n} of size n is calledalpha-Toeplitz if its entries obey the ruleA_{n}=[a_{r-alpha*s}]_{r,s=0}^{n-1}. Analogously, a matrix A_{n} again of sizen 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 andmore generally when dealing with multigrid/multilevel methods for structuredmatrices and approximations of boundary value problems. In this paper we studythe singular values of alpha-circulants and we provide an asymptotic analysisof the distribution results for the singular values of alpha-Toeplitz sequencesin the case where {a_{k}} can be interpreted as the sequence of Fouriercoeffcients of an integrable function f over the domain (-pi;pi). Somegeneralizations to the block, multilevel case, amounting to choose fmultivariate and matrix valued, are briefly considered.

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