A Fourier analysis for $(\theta,T)$-periodic functions and applications

TL;DR

This paper extends Fourier analysis to $( heta, T)$-periodic functions, establishing properties and inequalities, and applies these results to characterize the hypoellipticity and solvability of differential operators in this generalized setting.

Contribution

It introduces a Fourier analysis framework for $( heta, T)$-periodic functions and links operator properties on this class to classical periodic cases.

Findings

Established properties and inequalities for the Fourier transform of $( heta, T)$-periodic functions.

Characterized global hypoellipticity and solvability of operators on $( heta, T)$-periodic functions.

Connected the analysis of these operators to their classical periodic counterparts.

Abstract

We develop a Fourier analysis for a generalization of the class of periodic functions, often referred to as $(\theta, T)$-periodic functions, and prove several properties and inequalities related to the Fourier transform, including a type of Poincar\'e inequality, which extend the periodic case. As an application, we employ this analysis to show that a continuous linear operator acting on smooth $(\theta, T)$-periodic functions is globally hypoelliptic/solvable if and only if the corresponding operator which acts on periodic functions is globally hypoelliptic/solvable, and characterize the global hypoellipticity/solvability of a class of first order differential operators acting on the set of smooth $(\theta, T)$-periodic functions.