Eigenvectors of Toeplitz matrices from Fisher-Hartwig symbols with greater than, or equal to, one singularity

TL;DR

This paper analytically derives the asymptotic form of eigenvectors for Toeplitz matrices generated by Fisher-Hartwig symbols with multiple singularities, extending previous methods to more complex symbols.

Contribution

It extends the Wiener-Hopf method to derive eigenvectors for Fisher-Hartwig symbols with multiple singularities, beyond the single singularity case.

Findings

Derived asymptotic eigenvector forms for new Fisher-Hartwig symbols

Extended Wiener-Hopf method for complex symbols

Provided a table of eigenvalues for different functions

Abstract

Asymptotically, we analytically derive the form of eigenvectors for two Fisher-Hartwig symbols besides those which were previously investigated in a $2016$ work due to Movassagh and Kadanoff, in which the authors characterized the eigenpairs of Toeplitz matrices generated by Fisher-Hartwig symbols with one singularity. To perform such computations, we extend their methods which consists of formulating an eigenvalue problem, obtained by a Wiener-Hopf method, from which a suitable winding number is defined for passing to Fourier space, and introducing a factorization dependent upon the winding number, for other Fisher-Hartwig symbols which have previously been defined in the literature. Following the computations required for the proof after obtaining the asymptotic approximation of the eigenvectors, we provide a table from which eigenvalues of one Fisher-Hartwig symbol for different complex valued functions $b$ can be inferred.