Continuous frames, function spaces, and the discretization problem

Massimo Fornasier, Holger Rauhut

Research output: Contribution to journalArticlepeer-review

112 Scopus citations

Abstract

A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous superpositions. Associated to a given continuous frame we construct certain Banach spaces. Many classical function spaces can be identified as such spaces. We provide a general method to derive Banach frames and atomic decompositions for these Banach spaces by sampling the continuous frame. This is done by generalizing the coorbit space theory developed by Feichtinger and Gröchenig. As an important tool the concept of localization of frames is extended to continuous frames. As a byproduct we give a partial answer to the question raised by Ali, Antoine, and Gazeau whether any continuous frame admits a corresponding discrete realization generated by sampling.

Original languageEnglish
Pages (from-to)245-287
Number of pages43
JournalJournal of Fourier Analysis and Applications
Volume11
Issue number3
DOIs
StatePublished - Jun 2005
Externally publishedYes

Keywords

  • Atomic decompositions
  • Banach algebras of kernels
  • Banach frames
  • Continuous frames
  • Coorbit spaces
  • Discrete frames
  • Function spaces
  • General sampling methods
  • Localization of frames

Fingerprint

Dive into the research topics of 'Continuous frames, function spaces, and the discretization problem'. Together they form a unique fingerprint.

Cite this