Spectral Distribution in the Eigenvalues Sequence of Products of g-Toeplitz Structures

doi:10.4208/nmtma.OA-2017-0127

Spectral Distribution in the Eigenvalues Sequence of Products of g-Toeplitz Structures

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Year: 2019

Numerical Mathematics: Theory, Methods and Applications, Vol. 12 (2019), Iss. 3 : pp. 750–777

Abstract

Starting from the definition of an $n\times n$ $g$ -Toeplitz matrix, $\!T_{n,g}\!\!\left(\!u\!\right)\!=\!\left[\!\widehat{u}_{rgs}\!\right]_{\!r,s=0}^{n-1}\!,$ where $g$ is a given nonnegative parameter, $\{\widehat{u}_{k}\}$ is the sequence of Fourier coefficients of the Lebesgue integrable function $u$ defined over the domain $\mathbb{T}=(-\pi,\pi]$ , we consider the product of $g$ -Toeplitz sequences of matrices $\{T_{n,g}(f_{1})T_{n,g}(f_{2})\},$ which extends the product of Toeplitz structures $\{T_{n}(f_{1})T_{n}(f_{2})\},$ in the case where the symbols $f_{1},f_{2}\in L^{\infty}(\mathbb{T}).$ Under suitable assumptions, the spectral distribution in the eigenvalues sequence is completely characterized for the products of $g$ -Toeplitz structures. Specifically, for $g\geq2$ our result shows that the sequences $\{T_{n,g}(f_{1})T_{n,g}(f_{2})\}$ are clustered to zero. This extends the well-known result, which concerns the classical case (that is, $g=1$ ) of products of Toeplitz matrices. Finally, a large set of numerical examples confirming the theoretic analysis is presented and discussed.

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Journal Article Details

Publisher Name: Global Science Press

Language: English

DOI: https://doi.org/10.4208/nmtma.OA-2017-0127

Numerical Mathematics: Theory, Methods and Applications, Vol. 12 (2019), Iss. 3 : pp. 750–777

Published online: 2019-01

AMS Subject Headings:

Pages: 28

Keywords: Matrix sequences g-Toeplitz spectral distribution eigenvalues products of g-Toeplitz clustering.

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