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 - <p style="text-align: justify;">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.</p>