Year: 2025
Type: article
Abstract: We investigate the hydrodynamic behavior of zeroes of trigonometric polynomials under repeated differentiation. We show that if the zeroes of a real-rooted, degree d d trigonometric polynomial are distributed according to some probability measure ν \nu in the large d d limit, then the zeroes of its [ 2 t d ] [2td] -th derivative, where t > 0 t>0 is fixed, are distributed according to the free multiplicative convolution of ν \nu and the free unitary Poisson distribution with parameter t t . In the simplest special case, our result states that the zeroes of the [ 2 t d ] [2td] -th derivative of the trigonometric polynomial ( sin θ 2 ) 2 d (\sin \frac \theta 2)^{2d} (which can be thought of as the trigonometric analogue of the Laguerre polynomials) are distributed according to the free unitary Poisson distribution with parameter t t , in the large d d limit. The latter distribution is defined in terms of the function ζ = ζ t ( θ ) \zeta =\zeta _t(\theta ) which solves the implicit equation ζ − t tan
DOI: 10.1090/tran/9541
Author: Zakhar Kabluchko
Institution: University of Münster
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Deutsche Forschungsgemeinschaft
OpenAlex ID: W4414494885