## The Annals of Applied Probability

### Approximating mixed Hölder functions using random samples

Nicholas F. Marshall

#### 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
First available in Project Euclid: 18 October 2019

https://projecteuclid.org/euclid.aoap/1571385627

Digital Object Identifier
doi:10.1214/19-AAP1471

Mathematical Reviews number (MathSciNet)
MR4019880

#### 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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