On lacunary Toeplitz determinants

TL;DR

This paper develops Riemann-Hilbert problem techniques to derive the asymptotic expansion of lacunary Toeplitz determinants generated by holomorphic symbols, extending classical Szeg"o asymptotics with a new determinant-based correction.

Contribution

It introduces a novel Riemann-Hilbert approach to analyze lacunary Toeplitz determinants, providing explicit asymptotics including a determinant correction term.

Findings

Derived asymptotic expansion involving a determinant of size n+r

Extended Szeg"o asymptotics to lacunary cases

Applied Riemann-Hilbert techniques to Toeplitz determinants

Abstract

By using Riemann--Hilbert problem based techniques, we obtain the asymptotic expansion of lacunary Toeplitz determinants $\det_N\big[ c_{\ell_a-m_b}[f] \big]$ generated by holomorhpic symbols, where $\ell_a=a$ (resp. $m_b=b$) except for a finite subset of indices $a=h_1,\dots, h_n$ (resp. $b=t_1,\dots, t_r$). In addition to the usual Szeg\"{o} asymptotics, our answer involves a determinant of size $n+r$.