Resolving the Gibbs Phenomenon in Fractional Fourier Series via Inverse Polynomial Reconstruction

TL;DR

This paper extends the Inverse Polynomial Reconstruction Method to fractional Fourier series, effectively eliminating the Gibbs phenomenon for piecewise smooth functions and providing exponential convergence guarantees.

Contribution

It introduces a novel extension of IPRM to fractional Fourier series, addressing the Gibbs phenomenon in this generalized setting.

Findings

Complete elimination of the Gibbs phenomenon in numerical experiments

Exponential convergence for analytic functions

Conditioning governed by an angle-independent Gram matrix

Abstract

The fractional Fourier series generalizes the classical Fourier series by introducing a rotation angle $\alpha$ in the time-frequency plane, but inherits the Gibbs phenomenon for piecewise smooth functions. Unlike the classical setting, the chirp modulation factor renders the fractional partial sum complex-valued, corrupting both real and imaginary components simultaneously and making direct adaptation of classical remedies insufficient. The Inverse Polynomial Reconstruction Method (IPRM) resolves the Gibbs phenomenon by enforcing that the Fourier coefficients of a Gegenbauer polynomial expansion match the given spectral data, rather than projecting the corrupted partial sum onto a polynomial basis. This paper extends the IPRM to fractional Fourier series for the first time. The fractional transformation matrix is derived and its conditioning is shown to be governed by an $\alpha$-independent Gram matrix, which reveals the dependence on the Gegenbauer parameter $\lambda$ and the polynomial degree $m$, while being entirely insensitive to the transform angle. An $L^{\infty}$ error estimate is established, guaranteeing exponential convergence for analytic functions. Numerical experiments on piecewise analytic test functions demonstrate complete elimination of the Gibbs phenomenon and confirm the theoretical predictions.