Function approximation on arbitrary domains using Fourier extension frames

Fourier extension is an approximation scheme in which a function on an arbitary bounded domain is approximated using a classical Fourier series on a bounding box. On the smaller domain the Fourier series exhibits redundancy, and it has the mathematical structure of a frame rather than a basis. It is not trivial to construct approximations in this frame using function evaluations in points that belong to the domain only, but one way to do so is through a discrete least squares approximation. The corresponding system is extremely ill-conditioned, due to the redundancy in the frame, yet its solution via a regularized SVD is known to be accurate to very high (and nearly spectral) precision. Still, this computation requires O(N) operations. In this paper we describe an algorithm to compute such Fourier extension frame approximations in only O(N log N) operations for general 2D domains. The cost improves to O(N log N) operations for simpler tensor-product domains. The algorithm exploits a phenomenon called the plunge region in the analysis of time-frequency localization operators, which manifests itself here as a sudden drop in the singular values of the least squares matrix. It is known that the size of the plunge region scales like O(logN) in one dimensional problems. In this paper we show that for most 2D domains in the fully discrete case the plunge region scales like O(N logN), proving a discrete equivalent of a result that was conjectured by Widom for a related continuous problem. The complexity estimate depends on the Minkowski or box-counting dimension of the domain boundary, and as such it is larger than O(N logN) for domains with fractal shape.