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 formulae 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 a relatively modest number of additional function evaluations.