Mohammad Ayman Mursaleen

Paola Ferrari, Isabella Furci, Sean Hon et al.

Given a Lebesgue integrable function $f$ over $[0,2π]$, we consider the sequence of matrices $\{Y_nT_n[f]\}_n$, where $T_n[f]$ is the $n$-by-$n$ Toeplitz matrix generated by $f$ and $Y_n$ is the flip permutation matrix, also called the anti-identity matrix. Because of the unitary character of $Y_n$, the singular values of $T_n[f]$ and $Y_n T_n[f]$ coincide. However, the eigenvalues are affected substantially by the action of the matrix $Y_n$. Under the assumption that the Fourier coefficients are real, we prove that $\{Y_nT_n[f]\}_n$ is distributed in the eigenvalue sense as \[ ϕ_g(θ)=\left\{ \begin{array}{cc} g(θ), & θ\in [0,2π], -g(-θ), & θ\in [-2π,0), \end{array} \right.\, \] with $g(θ)=|f(θ)|$. We also consider the preconditioning introduced by Pestana and Wathen and, by using the same arguments, we prove that the preconditioned sequence is distributed in the eigenvalue sense as $ϕ_1$, under the mild assumption that $f$ is sparsely vanishing. We emphasize that the mathematical tools introduced in this setting have a general character and in fact can be potentially used in different contexts. A number of numerical experiments are provided and critically discussed.

Sean Hon, Mohammad Ayman Mursaleen, Stefano Serra-Capizzano

The singular value and spectral distribution of Toeplitz matrix sequences with Lebesgue integrable generating functions is well studied. Early results were provided in the classical Szeg{ő} theorem and the Avram-Parter theorem, in which the singular value symbol coincides with the generating function. More general versions of the theorem were later proved by Zamarashkin and Tyrtyshnikov, and Tilli. Considering (real) nonsymmetric Toeplitz matrix sequences, we first symmetrize them via a simple permutation matrix and then we show that the singular value and spectral distribution of the symmetrized matrix sequence can be obtained analytically, by using the notion of approximating class of sequences. In particular, under the assumption that the symbol is sparsely vanishing, we show that roughly half of the eigenvalues of the symmetrized Toeplitz matrix (i.e. a Hankel matrix) are negative/positive for sufficiently large dimension, i.e. the matrix sequence is symmetric (asymptotically) indefinite.