Duality for Frames

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 J Fourier Anal Appl (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.

Orthogonality of a pair of frames over locally compact abelian groups

“…Using Proposition 6.4, we make the following observation. One can find similar results on Gabor systems in different settings, for example, in [14,24,28,30,33,36,37], and various references within. Corollary 6.5.…”

“…In the last two decades, frame theory on LCA groups has become the focus of an active research, both in theory as well as in applications due to its potential to unify the continuous theory (integral representations) and the discrete theory (series expansions). Several researchers have made remarkable contributions in establishing the theory required to analyse frame properties on such groups (e.g., see [7,8,10,14,18,19,28,29,32,33,41]).…”

“…Among these systems, the study of frame properties such as duality of structured function systems (e.g., Gabor, wavelet, and shearlet systems) in different settings has got special attention due to their interesting theory and enormous applications in pure mathematics as well as in engineering areas such as signal processing, image processing etc. [3,10,14,15,18,30,37].…”

On Various R-duals and the Duality Principle

“…Therefore, S is the optimal upper bound in (8). In a similar way, it follows that 1/ S −1 is the optimal lower bound in (8).…”

Density and duality theorems for regular Gabor frames

“…We mention that duality results in the discrete case have been generalized in other directions; we refer the reader to [4,10,11,22,35] and the references therein. Generalizations of the density theorem also exist; in particular, Ramanathan and Steger [32] obtained density results for G (g, ∆) in L 2 (R n ), where ∆ is a discrete set, but not necessarily a subgroup.…”