NDLI: Duality for Frames

Content Provider	SpringerLink
Author	Fan, Zhitao Heinecke, Andreas Shen, Zuowei
Copyright Year	2015
Abstract	The subject of this article is the duality principle, which, well beyond its stand at the heart of Gabor analysis, is a universal principle in frame theory that gives insight into many phenomena. Its fiber matrix formulation for Gabor systems is the driving principle behind seemingly different results. We show how the classical duality identities, operator representations and constructions for dual Gabor frames are in fact aspects of the dual Gramian matrix fiberization and its sole duality principle, giving a unified view to all of them. We show that the same duality principle, via dual Gramian matrix analysis, holds for dual (or bi-) systems in abstract Hilbert spaces. The essence of the duality principle is the unitary equivalence of the frame operator and the Gramian of certain adjoint systems. An immediate consequence is, for example, that, even on this level of generality, dual frames are characterized in terms of biorthogonality relations of adjoint systems. We formulate the duality principle for irregular Gabor systems which have no structure whatsoever to the sampling of the shifts and modulations of the generating window. In case the shifts and modulations are sampled from lattices we show how the abstract matrices can be reduced to the simple structured fiber matrices of shift-invariant systems, thus arriving back in the well understood territory. Moreover, in the arena of multiresolution analysis (MRA)-wavelet frames, the mixed unitary extension principle can be viewed as the duality principle in a sequence space. This perspective leads to a construction scheme for dual wavelet frames which is strikingly simple in the sense that it only needs the completion of an invertible constant matrix. Under minimal conditions on the MRA, our construction guarantees the existence and easy constructability of non-separable multivariate dual MRA-wavelet frames. The wavelets have compact support and we show examples for multivariate interpolatory refinable functions. Finally, we generalize the duality principle to the case of transforms that are no longer defined by discrete systems, but may have discrete adjoint systems.
Ending Page	136
Page Count	66
Starting Page	71
File Format	PDF
ISSN	10695869
e-ISSN	15315851
Journal	Journal of Fourier Analysis and Applications
Issue Number	1
Volume Number	22
Language	English
Publisher	Springer US
Publisher Date	2015-05-15
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Dual frame Adjoint system Signal, Image and Speech Processing Completeness of sets of functions Wavelets and other special systems Abstract Harmonic Analysis Duality principle Wavelets Gabor frame Mathematical Methods in Physics Fourier Analysis Approximations and Expansions Filter bank General harmonic expansions, frames Dual Gramian analysis Partial Differential Equations
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis

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