Fourier Residual Networks Achieve Spectral Accuracy for Discontinuous Functions

TL;DR

This paper demonstrates that Fourier residual networks can spectrally approximate a wide class of discontinuous and smooth functions in one dimension, surpassing classical Fourier methods.

Contribution

It introduces a constructive framework showing Fourier residual networks' ability to achieve spectral convergence for diverse functions, including those with discontinuities.

Findings

Fourier residual networks achieve spectral convergence for discontinuous functions.

The approach overcomes limitations of classical Fourier approximation methods.

Numerical experiments validate the theoretical spectral approximation results.

Abstract

We present a constructive approximation framework for analyzing the expressive power of Fourier residual networks in approximating a broad class of one-dimensional functions. Our study covers both piecewise continuous functions -- including those with jump discontinuities in the function and its derivatives -- and fully smooth functions. We show that Fourier residual networks achieve spectral convergence without requiring periodicity or continuity, thereby overcoming key limitations of classical linear Fourier approximation and nonlinear methods, without being restricted to Barron-type function spaces. Our approach builds on classical techniques from approximation theory, including fixed-point iteration and Hermite interpolation by trigonometric polynomials. We support our theoretical results with numerical experiments based on both the constructed approximations and a randomized algorithm developed in our earlier work.