Convergence of fractional Fourier series on the torus and applications

TL;DR

This paper introduces fractional Fourier series on the fractional torus, exploring their properties, convergence, and applications to fractional PDEs and signal recovery, advancing the mathematical framework and practical utility of fractional Fourier analysis.

Contribution

It develops the theory of fractional Fourier series on the fractional torus, including convolution, approximation, and convergence, and applies these to fractional PDEs and signal processing.

Findings

Established pointwise convergence of fractional Fourier series.

Derived fractional Fourier inversion and Poisson summation formula.

Applied fractional Fourier series to solve fractional PDEs and recover signals.

Abstract

In this paper, we introduce the fractional Fourier series on the fractional torus and study some basic facts of fractional Fourier series, such as fractional convolution and fractional approximation. Meanwhile, fractional Fourier inversion and Poisson summation formula are also given. We further discuss the relationship between the decay of fractional Fourier coefficients and the smoothness of a function. Using the properties of fractional Fejer kernel, the pointwise convergence of fractional Fourier series can be established. Finally, we present the applications of fractional Fourier series to fractional partial differential equations with periodic boundary condition. Moreover, we apply approximation methods on the fractional torus to recover the non-stationary signals.