Continuous Frames, Function Spaces, and the Discretization Problem

By Massimo Fornasier, Holger Rauhut. 2004

162 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Mechanics, Geophysics

View Paper: https://www.semanticscholar.org/paper/c4b457ab8594655f9f52c597531b0924a5853c62

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Citations in the corpus

Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type (2010, methodology): “The Fourier transform is a bijection (in both cases) and its inverse is given by ϕ∨ = ϕ̂(−·).”
CONTINUOUS FRAMES IN HILBERT SPACES (2006, background): “For more details, the reader can refer to [1, 2, 3, 9, 12, 14, 17].”
The discretization problem for continuous frames (2016, methodology): “states, and chapter 16 of the book is devoted to the discretization problem. A solution for certain types of continuous frames was obtained by Fornasier and Rauhut using the theory of co-orbit spaces [FR]. We solve the discretization problem in its full generality with the following theorem which ch”
Generalized coorbit theory, Banach frames, and the relation to α‐modulation spaces (2008, methodology): “Therefore one aim of this paper is to generalize the approach in [9, 10] to more general operators Aσ; see also [23].”
Banach frames for α-modulation spaces (2007, methodology): “A detailed discussion on the relations between continuous and discrete characterization of α-modulation spaces is given in [11], in the context of recent generalizations of the coorbit space theory [12,13, 31 ].”
Coorbit space theory for quasi-Banach spaces (2005, unknown): “Recently, there has been some activity to provide generalizations to other settings than the classical one of integrable group representations [1, 2, 11 , 20, 21].”