Construction of Optimal Algorithms for Function Approximation in Gaussian Sobolev Spaces

This paper studies function approximation in Gaussian Sobolev spaces over the real line and measures the error in a Gaussian-weighted L^p-norm. We construct two linear approximation algorithms using n function evaluations that achieve the optimal or almost optimal rate of worst-case convergence in a Gaussian Sobolev space of order α. The first algorithm is based on scaled trigonometric interpolation and achieves the optimal rate n^-α up to a logarithmic factor. This algorithm can be constructed in almost-linear time with the fast Fourier transform. The second algorithm is more complicated, being based on spline smoothing, but attains the optimal rate n^-α.