Ten lectures on wavelets
SIAM Journal on Scientific Computing
Multigrid
Journal of Approximation Theory
A Note on Antireflective Boundary Conditions and Fast Deblurring Models
SIAM Journal on Scientific Computing
V-cycle Optimal Convergence for Certain (Multilevel) Structured Linear Systems
SIAM Journal on Matrix Analysis and Applications
Multigrid Methods for Multilevel Circulant Matrices
SIAM Journal on Scientific Computing
The asymptotic properties of the spectrum of nonsymmetrically perturbed Jacobi matrix sequences
Journal of Approximation Theory
Mass concentration in quasicommutators of Toeplitz matrices
Journal of Computational and Applied Mathematics
A note on the (regularizing) preconditioning of g-Toeplitz sequences via g-circulants
Journal of Computational and Applied Mathematics
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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.