Wavelet approximation of distributions with bounded variation derivatives
The topic of this paper is a wavelet approximation of distributions from the spaces \(\text{BV}^{r}\), which are the spaces of distributions \(f\) such that the distributional derivatives \(D^{\alpha} f\) with \(|\alpha |\leq r\) exist as measures of bounded variation. These spaces are a generalization of the classical \(\text{BV}\) spaces to higher orders of smoothness. The main aim of this paper is to generalize as many of known results about the Haar approximations as possible to higher orders of smoothness of wavelet approximations, where smoothness is measured in the sense of \(\text{BV}\). This paper discusses estimates for wavelet coefficients of \(\text{BV}^{r}\) distributions, direct (Jackson) and inverse (Bernstein) inequalities for \(n\)-term approximation of elements of \(\text{BV}^{r}\) in the \(L_{p}\) spaces using compactly supported wavelets. The optimal rates of approximation are established. The linear approximation in similar contexts is considered for comparison.
scientific article; zbMATH DE number 5562680
scientific article