The Annals of Applied Probability

Approximating mixed Hölder functions using random samples

Nicholas F. Marshall

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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


Export citation

References

  • [1] Bungartz, H.-J. and Griebel, M. (2004). Sparse grids. Acta Numer. 13 147–269.
    Zentralblatt MATH: 1118.65388
    Digital Object Identifier: doi:10.1017/S0962492904000182
  • [2] Gavish, M. and Coifman, R. R. (2012). Sampling, denoising and compression of matrices by coherent matrix organization. Appl. Comput. Harmon. Anal. 33 354–369.
    Mathematical Reviews (MathSciNet): MR2950134
    Zentralblatt MATH: 1256.65036
    Digital Object Identifier: doi:10.1016/j.acha.2012.02.001
  • [3] Griebel, M. and Hamaekers, J. (2014). Fast discrete Fourier transform on generalized sparse grids. In Sparse Grids and Applications—Munich 2012. Lecture Notes in Computational Science and Engineering 97 75–107. Springer, Cham.
    Zentralblatt MATH: 1316.65119
  • [4] Needell, D. (2010). Randomized Kaczmarz solver for noisy linear systems. BIT 50 395–403.
    Zentralblatt MATH: 1195.65038
    Digital Object Identifier: doi:10.1007/s10543-010-0265-5
  • [5] Smoljak, S. A. (1963). Quadrature and interpolation formulae on tensor products of certain function classes. Dokl. Akad. Nauk SSSR 148 1042–1045.
  • [6] Strohmer, T. and Vershynin, R. (2009). A randomized Kaczmarz algorithm with exponential convergence. J. Fourier Anal. Appl. 15 262–278.
    Zentralblatt MATH: 1169.68052
    Digital Object Identifier: doi:10.1007/s00041-008-9030-4
  • [7] Strömberg, J.-O. (1998). Computation with wavelets in higher dimensions. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998) 523–532.

The Institute of Mathematical Statistics

The Annals of Applied Probability

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more