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Constructive sparse approximation with respect to the Faber-Schauder system
发布时间: 2018-12-03     11:02   【返回上一页】 发布人:Glenn Byrenheid


函数空间及其应用讨论班

 

 

 

 : Constructive sparse approximation with respect to the Faber-Schauder system

 

报告人: Dr. Glenn Byrenheid (Friedrich Schiller University Jena, Germany)     

 

 :  12月4日16:10至17:10

 

 :  教二楼405室

 

We consider approximations of multivariate functions using $m$ terms from its tensorized Faber-Schauder expansion. The univariate Faber-Schauder system on $[0,1]$ is given by dyadic dilates and translates (in the wavelet sense) of the $L_{\infty}$ normalized simple hat functions with support in [0,1]. We obtain a hierarchical basis which will be tensorized over all levels (hyperbolic) to get the dictionary $\mathcal{F}$. The worst-case error with respect to a class of functions $\mathbb{F}\hookrightarrow X$ is measured by the usual best $m$-term widths denoted by $\sigma_m(\mathbb{F},\mathcal{F})_X$, where the error is measured in $X$. We constructively prove the following sharp asymptotical bound for the class of Besov spaces with small mixed smoothness (i.e. $1/p<r<\min\{1/\theta-1,2\}$) in $L_q$ $(p<q\leq \infty)$ $\sigma_m(S^r_{p,\theta} B,\mathcal{F})_q\asymp m^{-r}.$
Note, that this asymptotical rate of convergence does not depend on the dimension $d$ (only the constants behind). In addition, this result holds for $q=\infty$ and to our best knowledge this is one of the first sharp results involving $L_{\infty}$ as a target space for sparse approximation in the context of spaces with dominating mixed smoothness. We emphasize two more things. First, the selection procedure for the coefficients is a level-wise constructive greedy strategy which only touches a finite prescribed number of coefficients. And second, due to the use of the Faber-Schauder system, the coefficients are finite linear combinations of discrete functions values. Hence, this method can be considered as a nonlinear adaptive sampling algorithm leading to a pure polynomial rate of convergence for any d.