Multivariate modified Fourier expansions

In this paper, we review recent advances in the approximation of multivariate functions using eigenfunctions of the Laplace operator subject to homogeneous Neumann boundary conditions. Such eigenfunctions are known explicitly on a variety of domains, including the d-variate cube, equilateral triangle and numerous other higher dimensional simplices. Practical construction of truncated expansions is achieved using a mixture of asymptotic and classical quadratures. Moreover, by exploiting the hyperbolic cross, the number of expansion coefficients need only grow mildly with dimension. Despite converging uniformly throughout the domain, the rate of convergence of such expansions may be slow. We review two techniques to accelerate convergence. The first smoothes the function by interpolating certain derivatives of the function evaluated on the boundary of the domain. The second numerically computes a smooth, periodic extension of the function on a larger domain.