From high oscillation to rapid approximation III: Multivariate expansions

In this paper we expand upon the theme of modified Fourier expansions and extend the theory to a multivariate setting and to expansions in eigenfunctions of the Laplace– Neumann operator. We pay detailed attention to expansions in a d-dimensional cube and to an effective derivation of expansion coefficients there by means of quadratures of highly oscillatory integrals. Thus, we present asymptotic and Filon-type formulæ for an effective derivation of expansion coefficients and discuss their design and relative advantages. Such methods are effective only for large indices, hence we introduce and analyse alternative quadrature schemes that require relatively modest number of additional function evaluations.