On Bivariate Smoothness Spaces Associated with Nonlinear Approximation
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- 作者：S. Dekel ; D. Leviatan and M. Sharir
- 关键词：Multivariate nonlinear approximation ; Smoothness spaces ; Besov spaces ; Modulus ofsmoothness ; K ; Functional ; Mumford– ; Shah functional ; Piecewise polynomials approximation ; Wavelets
- 刊名：Constructive Approximation
- 出版时间：August 2004
- 出版年：2004
- 期刊代码：34_0176-4276
- 类别：bs
- 卷：20
- 期：4
- 页码：625-646
- 数据来源：sp
摘要
In recent years there have been various attempts at therepresentations of {\mbox multivariate} signals such as images, whichoutperform wavelets. As is well known, wavelets are not optimal inthat they do not take full advantage of the geometricalregularities and singularities of the images. Thus theseapproaches have been based on tracing curves of singularities andapplying bandlets, curvelets, ridgelets, etc., or allocating some weights to curves ofsingularities like the Mumford–Shah functional and itsmodifications. In the latter approach a function is approximatedon subdomains where it is smoother but there is a penalty in theform of the total length (or other measurement) of thepartitioning curves. We introduce a combined measure of smoothnessof the function in several dimensions by augmenting its smoothnesson subdomains by the smoothness of the partitioning curves.Also, it is known that classical smoothness spaces fail tocharacterize approximation spaces corresponding to multivariatepiecewise polynomial nonlinear approximation. We show how theproposed notion of smoothness can almost characterize thesespaces. The question whether the characterization proposed in thiswork can be further simplified remains open.