For a given nonnegative integer $g$, a matrix $A_n$ of size $n$ is called $g$-Toeplitz if its entries obey the rule $A_n=\left[a_{r-g s}\right]_{r,s=0}^{n-1}$. Analogously, a matrix $A_n$ again of size $n$ is called $g$-circulant if $A_n=\left[a_{(r-g s)\ {\rm mod}\, n}\right]_{r,s=0}^{n-1}$. Such kind of 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 we 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.
Spectral features and asymptotic properties for g-Circulants and g-Toeplitz sequences
SERRA CAPIZZANO, STEFANO;
2010-01-01
Abstract
For a given nonnegative integer $g$, a matrix $A_n$ of size $n$ is called $g$-Toeplitz if its entries obey the rule $A_n=\left[a_{r-g s}\right]_{r,s=0}^{n-1}$. Analogously, a matrix $A_n$ again of size $n$ is called $g$-circulant if $A_n=\left[a_{(r-g s)\ {\rm mod}\, n}\right]_{r,s=0}^{n-1}$. Such kind of 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 we 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.
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