Sparse generalized Fourier series via collocation-based optimization

摘要

Generalized Fourier series with orthogonal polynomial bases have usefulapplications in several fields, including differential equations, patternrecognition, and image and signal processing. However, computing thegeneralized Fourier series can be a challenging problem even for relativelywell behaved functions. In this paper a method for approximating a sparsecollection of Fourier-like coefficients is presented that uses a collocationtechnique combined with an optimization problem inspired by recent results incompressed sensing research. The discussion includes approximation error ratesand numerical examples to illustrate the effectiveness of the method. Oneexample displays the accuracy of the generalized Fourier series approximationfor several test functions, while the other is an application of thegeneralized Fourier series approximation to rotation-invariant patternrecognition in images.