1Rapid function approximation by modified Fourier series

Olver We review a set of algorithms and techniques to approximate smooth functions on a domain Ω ⊂ Rd by an expansion in eigenfunctions of the Laplacian. We refer to such expansions as modified Fourier series. These series converge pointwise everywhere in the domain of approxima-tion, including on the boundary, at an algebraic rate that is essentially arbitrary. The computational complexity of the transformation is only linear in the number of terms of the expansion. Moreover, additional terms can be computed adaptively and efficiently. 1.