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Volume 12, Issue 3
Spectral Distribution in the Eigenvalues Sequence of Products of g-Toeplitz Structures

Eric Ngondiep

Numer. Math. Theor. Meth. Appl., 12 (2019), pp. 750-777.

Published online: 2019-04

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  • 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.

  • AMS Subject Headings

65F10, 15A18, 47B36, 47B65

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-12-750, author = {}, title = {Spectral Distribution in the Eigenvalues Sequence of Products of g-Toeplitz Structures}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2019}, volume = {12}, number = {3}, pages = {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.

}, issn = {2079-7338}, doi = {https://doi.org/ 10.4208/nmtma.OA-2017-0127}, url = {http://global-sci.org/intro/article_detail/nmtma/13129.html} }
TY - JOUR T1 - Spectral Distribution in the Eigenvalues Sequence of Products of g-Toeplitz Structures JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 750 EP - 777 PY - 2019 DA - 2019/04 SN - 12 DO - http://doi.org/ 10.4208/nmtma.OA-2017-0127 UR - https://global-sci.org/intro/article_detail/nmtma/13129.html KW - Matrix sequences, g-Toeplitz, spectral distribution, eigenvalues, products of g-Toeplitz, clustering. AB -

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.

Eric Ngondiep. (2019). Spectral Distribution in the Eigenvalues Sequence of Products of g-Toeplitz Structures. Numerical Mathematics: Theory, Methods and Applications. 12 (3). 750-777. doi: 10.4208/nmtma.OA-2017-0127
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