Question

I am looking for a good starting point (book or articles) for studying Toeplitz matrices. Specifically as mentioned in the title, I am most interested in the case where they are of the form $$A = {\phi(i-j)}_{i,j\in\mathbb{Z}}$$ where $\phi:\mathbb{R}\to\mathbb{R}$ and $\phi(i-j)=\phi(j-i)$.

I so far have looked at "Analysis of Toeplitz Operators" by Bottcher and Silbermann, but wonder if there might be some more references that address my specific interest, and if anything can be said about the explicit form of the inverse.

I apologize if this has been answered before, but I did not find much in searching old posts.

Answer 1

I strongly recommend the following book

Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators Lloyd N. Trefethen & Mark Embree

The first chapter is partly devoted to Toeplitz matrices, although their interest is focused on the non-selfadjoint case. Anyhow, the book is pleasant to read and very informative.