Quarkonial frames with compression properties
type: Item
instance of: scholarly article
title: Quarkonial frames with compression properties
zbMATH Open document ID: 1383.46025
author: Stephan Dahlke
author: Philipp Keding
author: Thorsten Raasch
published in: Calcolo
publication date: 2017-10-27T00:00:00Z
review text: Let \(\sum_{k\in Z} \phi (x-k) =1\), \(x\in\mathbb R\), be a partition of unity in \(\mathbb R\), where \(\phi\) is a smooth compactly supported function. Let \(\phi_p (x) = x^p \phi (x)\), \(p\in \mathbb N_0\). The paper deals with the quarks or quarklets \[ \phi_{p,j,k} (x) = 2^{j/2} \phi_p \big( 2^j x-k), \qquad p,j \in \mathbb N_0, \quad k\in \mathbb Z, \quad x\in \mathbb R. \] These basic functions are highly redundant compared with the usual wavelet bases. But they are easy to construct and highly flexible. In addition, they are frames in \(L_2 ( \mathbb R)\) and, in particular, in the (fractional) Sobolev spaces \(H^s ( \mathbb R)\), \(s>0\). Section 3 deals with related estimates of Jackson and Bernstein type. The frame property is studied in Section 4 (frames in \(L_2 ( \mathbb R)\)) and Section 5 (frames in \(H^s ( \mathbb R)\), \(s>0\)). This is applied in Section 6 to some elliptic problems.
reviewed by: Hans Triebel
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 65F50
zbMATH DE Number: 6798984
zbMATH Keywords: quarkonial decompositions
zbMATH Keywords: frames
zbMATH Keywords: partition-of-unity methods
zbMATH Keywords: Jackson and Bernstein inequalities
zbMATH Keywords: compression
zbMATH Keywords: wavelets
describes a project that uses: PUMA
MaRDI profile type: Publication
full work available at URL: s10092-016-0210-3
OpenAlex ID: W2562555151
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DOI: 10.1007/S10092-016-0210-3
description: scientific article; zbMATH DE number 6798984