The theory of reproducing systems on locally compact abelian groups

Gitta Kutyniok¹ ; Demetrio Labate²

¹Institute of Mathematics Justus-Liebig-University Giessen Arndtstr. 2 35392 Giessen, Germany
²Department of Mathematics North Carolina State University Campus Box 8205 Raleigh, NC 27695, U.S.A.

Colloquium Mathematicum, Tome 106 (2006) no. 2, pp. 197-220

Résumé

A reproducing system is a countable collection of functions $\{\phi_j: j \in {\cal J}\}$ such that a general function $f$ can be decomposed as $f = \sum_{j \in {\cal J}} c_j(f) \, \phi_j$, with some control on the analyzing coefficients $c_j(f)$. Several such systems have been introduced very successfully in mathematics and its applications. We present a unified viewpoint in the study of reproducing systems on locally compact abelian groups $G$. This approach gives a novel characterization of the Parseval frame generators for a very general class of reproducing systems on $L^2(G)$. As an application, we obtain a new characterization of Parseval frame generators for Gabor and affine systems on $L^2(G)$.

Détail

DOI : 10.4064/cm106-2-3

Keywords: reproducing system countable collection functions phi cal general function decomposed sum cal phi control analyzing coefficients several systems have introduced successfully mathematics its applications present unified viewpoint study reproducing systems locally compact abelian groups nbsp approach gives novel characterization parseval frame generators general class reproducing systems application obtain characterization parseval frame generators gabor affine systems