The Annals of Applied Probability

Approximating mixed Hölder functions using random samples

Nicholas F. Marshall

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Abstract

Suppose $f:[0,1]^{2}\rightarrow \mathbb{R}$ is a $(c,\alpha )$-mixed Hölder function that we sample at $l$ points $X_{1},\ldots ,X_{l}$ chosen uniformly at random from the unit square. Let the location of these points and the function values $f(X_{1}),\ldots ,f(X_{l})$ be given. If $l\ge c_{1}n\log^{2}n$, then we can compute an approximation $\tilde{f}$ such that \begin{equation*}\|f-\tilde{f}\|_{L^{2}}=\mathcal{O}\big(n^{-\alpha}\log^{3/2}n\big),\end{equation*} with probability at least $1-n^{2-c_{1}}$, where the implicit constant only depends on the constants $c>0$ and $c_{1}>0$.

Article information

Source
Ann. Appl. Probab., Volume 29, Number 5 (2019), 2988-3005.

Dates
Received: October 2018
First available in Project Euclid: 18 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1571385627

Digital Object Identifier
doi:10.1214/19-AAP1471

Mathematical Reviews number (MathSciNet)
MR4019880

Subjects
Primary: 26B35: Special properties of functions of several variables, Hölder conditions, etc.
Secondary: 60G42: Martingales with discrete parameter 42B35: Function spaces arising in harmonic analysis

Keywords
Hölder condition sparse grids randomized Kaczmarz

Citation

Marshall, Nicholas F. Approximating mixed Hölder functions using random samples. Ann. Appl. Probab. 29 (2019), no. 5, 2988--3005. doi:10.1214/19-AAP1471. https://projecteuclid.org/euclid.aoap/1571385627


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