English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Given a Lebesgue integrable function $f$ over $[0,2\pi]$, we consider thesequence 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, alsocalled the anti-identity matrix. Because of the unitary character of $Y_n$, thesingular values of $T_n[f]$ and $Y_n T_n[f]$ coincide. However, the eigenvaluesare affected substantially by the action of the matrix $Y_n$. Under theassumption that the Fourier coefficients are real, we prove that$\{Y_nT_n[f]\}_n$ is distributed in the eigenvalue sense as  \[  \phi_g(\theta)=\left\{  \begin{array}{cc}  g(\theta), & \theta\in [0,2\pi],  -g(-\theta), & \theta\in [-2\pi,0),  \end{array}  \right.\,  \] with $g(\theta)=|f(\theta)|$. We also consider the preconditioningintroduced by Pestana and Wathen and, by using the same arguments, we provethat the preconditioned sequence is distributed in the eigenvalue sense as$\phi_1$, under the mild assumption that $f$ is sparsely vanishing. Weemphasize that the mathematical tools introduced in this setting have a generalcharacter and in fact can be potentially used in different contexts. A numberof numerical experiments are provided and critically discussed.

The eigenvalue distribution of special $2$-by-$2$ block matrix sequences, with applications to the case of symmetrized Toeplitz structures