A hybrid reconstruction of piece-wise smooth functions from non-uniform Fourier data

TL;DR

This paper introduces a hybrid method combining non-uniform Fourier data filtering and stable extrapolation to accurately reconstruct piece-wise smooth functions with jump discontinuities, achieving exponential accuracy.

Contribution

It extends the filter method to non-uniform Fourier data and combines it with extrapolation to improve convergence near discontinuities.

Findings

Exponential convergence away from jump discontinuities.

Slower convergence near jumps due to Gibbs phenomenon.

Hybrid method achieves uniform exponential accuracy.

Abstract

In this paper, we consider the problem of reconstructing piece-wise smooth functions from their non-uniform Fourier data. We first extend the filter method for uniform Fourier data to the non-uniform setting by using the techniques of admissible frames. We show that the proposed non-uniform filter method converges exponentially away from the jump discontinuities. However, the convergence rate is significantly slower near the jump discontinuities due to the Gibbs phenomenon. To overcome this issue, we combine the non-uniform filter method with a stable extrapolation method to recover the function values near the jump discontinuities. We show that the proposed hybrid method could achieve exponential accuracy uniformly on the entire domain. Numerical experiments are provided to demonstrate the performance of the proposed method.