Construction of Optimal Algorithms for Function Approximation in Gaussian Sobolev Spaces

TL;DR

The paper tackles optimal linear function-approximation in Gaussian-weighted Sobolev spaces on the real line by constructing two explicit algorithms: (i) a truncated-domain scaled trigonometric interpolation method $A_n^{\dagger}$ that leverages FFT for near-optimal rates (up to a log factor) and (ii) a spline-smoothing based method $A_n^{*}$ that achieves the exact optimal rate $n^{-\alpha}$ (with higher cost). The analysis introduces a decay-aware framework for truncation, periodization, and auxiliary periodic functions to bound interpolation errors, and a generic interval-wise spline approach using Matérn kernels to realize the optimal rate for the case $q=2$, with extensions and robustness considerations discussed. The results connect to prior literature on Hermite-type spaces, quadrature, and randomized versus deterministic settings, and discuss higher-dimensional extensions via tensor-product or sparse-grid constructions. Practically, the work provides implementable strategies with explicit cost bounds ($\mathcal{O}(n\log n)$ for the FFT-based method) and clear guidance on parameter choices for achieving near-optimal or optimal convergence in Gaussian Sobolev sampling.

Abstract

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 $n^{-α}$.

Figures (2)

• Figure 1: Comparison of the evaluation points used by the algorithms $A^\dagger_n$ and $A^*_n$ from Sections \ref{['sec:trig-interpolation']} and \ref{['sec:optimal']}, respectively, for $n = 30$ and $n = 62$. The uniform points used by $A^\dagger_n$ in \ref{['eq:Aa-def']}, with $T$ given in \ref{['eq:T-selection']} for $\alpha = 4$, $q=2$, $p=1$ and $\varepsilon = 0.25$, are displayed in the top panels. The bottom panels display one possible set of points that can be used to construct the algorithm $A^*_n$ of Corollary \ref{['cor:spline-convergence']}. The density of these points decreases exponentially fast when moving away from the origin. The algorithm $A^*_n$ converges with the optimal rate $n^{-\alpha}$, while the rate of convergence of $A^\dagger_n$ is optimal up to a logarithmic factor.
• Figure 2: A comparison of trigonometric interpolation from Section \ref{['sec:trig-interpolation']} and spline approximation from Section \ref{['sec:optimal']} when $f(x) = \lvert x \rvert$, $p=1$ and $q=2$. We plot the weighted approximations $\rho A^\dagger_n(f)$ with $n=31$ and $\varepsilon=0.25$ (left) and $\rho A^*_n(f)$ with $n=30$ (right). The spline approximation has been constructed using a Matérn kernel of order $\gamma = 3/2$ [see \ref{['eq:matern-kernel']}], which is a reproducing kernel for a Sobolev space of order two. Observe how the spline approximation is more accurate close to the origin than the trigonometric interpolant.

Theorems & Definitions (32)

• definition 2.1: Trigonometric interpolation with cutoff $A^\dagger_n$
• lemma 2.2: Decay of the function $g=f\rho^{1/p}$
• proof
• lemma 2.3: Auxiliary periodic function and its properties
• proof
• lemma 2.4: Norm estimate on bounded intervals
• proof
• lemma 2.5: Boundedness of $\|g\circ S_T\|_{W^{\alpha,q}([0,2\pi])}$
• proof
• lemma 2.6: $L^p$ interpolation error for periodic functions