An Isaac Newton Institute Workshop

Effective Computational Methods for Highly Oscillatory Problems: The Interplay between Mathematical Theory and Applications

Modified Fourier expansions and spectral problems for highly oscillatory Fredholm operators

Authors: H. Brunner (Memorial University), A. Iserles (University of Cambridge), S. Nørsett (NTNU Trondheim)

Abstract

Although highly oscillatory Fredholm operators are compact and have point spectrum, their calculation by standard means, e.g. the finite section method, is notoriously difficult. As an alternative, we propose expanding the underlying eigenfunctions in modified Fourier series. This leads to infinite-dimensional algebraic eigenvalue problems that exhibit intriguing structure and rapid decay of coefficients. This is exploited in an effective numerical algorithm.