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Construction of Optimal Algorithms for Function Approximation in Gaussian Sobolev Spaces (2402.02917v1)
Published 5 Feb 2024 in math.NA and cs.NA
Abstract: This paper studies function approximation in Gaussian Sobolev spaces over the real line and measures the error in a Gaussian-weighted $Lp$-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 $\alpha$. The first algorithm is based on scaled trigonometric interpolation and achieves the optimal rate $n{-\alpha}$ 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 $n{-\alpha}$.