# Duality

Duality Analysis

The development of the duality analysis started from  and was completed in . The concept of the dual Gramian matrix and its analysis are introduced and a tool of fiberization was invented for the shift-invariant system in . The dual Gramian analysis and fiberization were further developed for generalized shift-invariant systems in . The theory developed in  was then applied to the Gabor systems and wavelet systems.

The duality analysis was further developed in [1,2].   In [1, 2], we discovered 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. The essence of the duality principle is the unitary equivalence of the dual Gramian matrix of a frame system and the Gramian matrix 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. 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 of [3,4].

The duality analysis provide a comprehensive understanding and lay a foundation of Gabor frames, wavelet frames, and frames in Hilbert spaces, and their applications.

1. Zhitao Fan, Andreas Heinecke, Zuowei Shen, Duality for frames, Journal of Fourier Analysis and Applications, 22(1), (2016), 71-136. PDF
2. Zhitao Fan, Hui Ji, Zuowei Shen, Dual Gramian analysis: duality principle and unitary extension principle, Mathematics of Computation, 85 (2016) 239-270. PDF
3. Amos Ron, Zuowei Shen, Generalized shift invariant systems, Constructive Approximation, 22 (2005), 1-45. gs.pdf
4. Amos Ron, Zuowei Shen, Frames and stable bases for shift invariant subspaces of L2(Rd), Canadian Journal of Mathematics, 47(1995), 1051-1094. PDF

Duality Principle

Since the Gabor system is shift-invariant, the dual Gramian analysis developed for shift-invariant systems was applied in  to obtain the duality principle for Gabor frame systems. The key idea was the introduction of an adjoint system of a given Gabor system. The adjoint system is also a Gabor system.  The essence of the duality principle is the unitary equivalence of the dual Gramian matrix of a frame system and the Gramian matrix of certain adjoint systems. Hence, the frame property of a Gabor system is characterized by the Riesz property of its adjoint system. With the development of the duality analysis, the duality principle was generalized to frames in Hilbert space and became the core principle of frame theory. With this, we formulated the duality principle for irregular Gabor systems which have no structure whatsoever to the sampling of the shifts and modulations of the generating window.

1. Amos Ron, Zuowei Shen, Weyl-Heisenberg frames and Riesz bases in \$L_2(R^d)\$, Duke Mathematical Journal, 89 (1997), 237-282. PDF
2. Amos Ron, Zuowei Shen, Frames and stable bases for subspaces of L2(Rd): the duality principle of Weyl-Heisenberg sets, Proceedings of the Lanczos Centenary Conference Raleigh, NC, M. Chu, R. Plemmons, D. Brown, and D. Ellison eds., SIAM Pub. (1993) 422-425. duality.pdf

Extension Principles
The wavelet system is not shift-invariant. In , we completely unraveled the structure of the wavelet system with the aid of two new notions: the affine product, and a quasi-affine system in . This led to a characterization of wavelet frames; the induced characterization of tight wavelet frames was in terms of exact orthogonality relations that the wavelets should satisfy on the Fourier domain. The affine product can be factored in during a multiresolution analysis construction, and this led to a complete characterization of all tight frames that can be constructed by such methods. Moreover, this characterization led to the unitary extension principle.

The unitary extension principle was further generalized to the mixed extension principle in  and the oblique extension principle in . Moreover, the unitary extension principle can be viewed as the duality principle in sequence space as shown in the duality analysis of frames in Hilbert space.  This perspective led to a construction scheme for wavelet frames which is strikingly simple in the sense that it only needs the completion of an invertible constant matrix which makes the construction of wavelet frame straightforward. The duality analysis and extension principles led to many interesting constructions of wavelet frames beyond those given here.

1. Ingrid Daubechies, Bin Han, Amos Ron, Zuowei Shen, Framelets:MRA-based constructions of wavelet frames, Applied and Computational Harmonic Analysis, 14 (2003), 1-46. dhrs.pdf
2. Amos Ron, Zuowei Shen, Affine systems in L2(Rd): the analysis of the analysis operator, Journal of Functional Analysis, 148 (1997), 408-447. PDF
3. Amos Ron, Zuowei Shen, Affine systems in L2(Rd): dual systems, Journal of Fourier Analysis and Applications, 3 (1997), 617-637. PDF