Recovery operator for Haar wavelet spaces

Abstract

In this paper we present a constructive approximation of a function by using its values at certain points. We consider the approximation of functions, the their Fourier — Haar coefficients of which decay rapidly.

We provide a method for calculating the Fourier — Haar coefficients and analyse the convergence rate of the error. If the function is a Haar polynomial with a degree at most $2^{m}-1$ (meaning that $\prod_{i=1}^dk_i<2^{m}$, where $k_i$ is a degree of $i$th variable), then the method approximates Fourier — Haar coefficients with the error $O(m^{d-1}2^m)$, where $d$ is the amount of points used for the approximation. The method is based on a sparse grid with weights and nodes given explicitly.

We construct a recovery operator based on our method. The recovery operator is a Haar polynomial with coefficients expressed in terms of the values of function. The recovery operator is exact for the corresponding Haar polynomial. We analyse the error made by the recovery operator in the metric of $L_{\infty}$ and obtain an upper bound for the error in terms of $m$.