The Annals of Applied Probability

Approximating mixed Hölder functions using random samples

Nicholas F. Marshall

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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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

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

Hölder condition sparse grids randomized Kaczmarz


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

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