Generalized gaussian bounds for discrete convolution powers

TL;DR

This paper establishes a generalized Gaussian bound for powers of discrete convolution operators in one dimension, extending previous results to broader classes of Fourier transforms with finitely many maxima.

Contribution

It introduces a uniform bound applicable to convolution operators with Fourier transforms that are rational functions, expanding beyond trigonometric polynomial cases.

Findings

Derived a generalized Gaussian bound for convolution powers

Extended previous results to rational function Fourier transforms

Allowed maxima at finitely many points in the Fourier domain

Abstract

We prove a uniform generalized gaussian bound for the powers of a discrete convolution operator in one space dimension. Our bound is derived under the assumption that the Fourier transform of the coefficients of the convolution operator is a trigonometric rational function, which generalizes previous results that were restricted to trigonometric polynomials. We also allow the modulus of the Fourier transform to attain its maximum at finitely many points over a period.