Sparse generalized Fourier series via collocation-based optimization

TL;DR

This paper introduces a collocation-based optimization method for efficiently computing sparse generalized Fourier series coefficients, enhancing approximation accuracy in applications like differential equations and pattern recognition.

Contribution

It presents a novel approach combining collocation and compressed sensing techniques to approximate sparse Fourier-like coefficients more effectively.

Findings

Effective approximation of test functions demonstrated

Application to rotation-invariant pattern recognition shown

Error rates analyzed and discussed

Abstract

Generalized Fourier series with orthogonal polynomial bases have useful applications in several fields, including differential equations, pattern recognition, and image and signal processing. However, computing the generalized Fourier series can be a challenging problem even for relatively well behaved functions. In this paper a method for approximating a sparse collection of Fourier-like coefficients is presented that uses a collocation technique combined with an optimization problem inspired by recent results in compressed sensing research. The discussion includes approximation error rates and numerical examples to illustrate the effectiveness of the method. One example displays the accuracy of the generalized Fourier series approximation for several test functions, while the other is an application of the generalized Fourier series approximation to rotation-invariant pattern recognition in images.