Construction of optimal algorithms for function approximation in Gaussian Sobolev spaces

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2026年3月30日

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Yuya Suzuki,
Toni Karvonen

摘要

This paper studies function approximation in Gaussian Sobolev spaces over the real line and measures the error in a Gaussian-weighted $L^p$ $L^{p}$ -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 $\alpha $ $α$ . The first algorithm is based on scaled trigonometric interpolation and achieves the optimal rate $n^{-\alpha }$ $n^{- α}$ 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 $n^{-\alpha }$ $n^{- α}$ .