Arieh Iserles.

Title: Highly oscillatory Fredholm operators: from spectral methods to 
modified Fourier expansions

Abstract: In this talk we report recent advances in the calculation of
spectra of complex-valued highly oscillatory Fredholm operators by the
finite section method. Standard considerations based on spectral
methods seem to indicate that expansions in Legendre polynomials are
likely to lead to rapid convergence, hence to small matrices. However,
calculation of matrix coefficients is expensive. On the other hand,
modified Fourier expansions come with rapid algorithms for the
calculation of coefficients. Moreover, their slower convergence is
offset by the technique of hyperbolic cross. So far, all is intuitive
-- but numerical results seem to indicate that, surprisingly, the
"slowly convergent" modified Fourier basis actually leads to smaller
matrices. Careful asymptotic analysis reveals the truth: numerical
results are right and intuition wrong!