Duality

Duality Analysis
This project was initiated with Amos Ron. The development of duality analysis began in [4] and was completed in [1]. We introduced the dual Gramian matrix and invented a fiberization tool for shift-invariant systems in [4], generalized shift-invariant systems in [3], and applied it to Gabor systems and wavelet systems. The duality analysis was further developed in [1, 2], where we discovered that the duality principle is a universal principle in frame theory that provides 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 its adjoint system. As a result, the dual frame property is characterized by the biorthogonality property of adjoint systems. When the shifts and modulations are on lattices, we show how the abstract matrices can be reduced to the simple structured fiber matrices of shift-invariant systems, thus returning to the well-understood territory of [3, 4].

The duality analysis provides a comprehensive understanding and lays the foundation for 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, we can apply the dual Gramian analysis for shift-invariant systems [1] to obtain the duality principle for Gabor frame systems. The discovery of the adjoint system is essential because of the unitary equivalence between the dual Gramian matrix of a frame system and the Gramian matrix of its adjoint system. Hence, we can use the Reisz property of the adjoint system to characterize the frame property of the Gabor system. The duality principle has been generalized to frames in Hilbert space and has become the core principle of frame theory. Furthermore, we have formulated the duality principle for irregular Gabor systems without a lattice structure 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 [2], 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 [2]. 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 [3] and the oblique extension principle in [1]. 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