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Dual Gramian analysis: Duality principle and unitary extension principle


Authors: Zhitao Fan, Hui Ji and Zuowei Shen
Journal: Math. Comp. 85 (2016), 239-270
MSC (2010): Primary 42C15; Secondary 42C40, 42C30, 65T60
Published electronically: June 23, 2015
MathSciNet review: 3404449
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Abstract: Dual Gramian analysis is one of the fundamental tools developed in a series of papers by Amos Ron and Zouwei Shen for studying frames. Using dual Gramian analysis, the frame operator can be represented as a family of matrices composed of the Fourier transforms of the generators of (generalized) shift-invariant systems, which allows us to characterize most properties of frames and tight frames in terms of their generators. Such a characterization is applied in the above-mentioned papers to two widely used frame systems, namely Gabor and wavelet frame systems. Among many results, we mention here the discovery of the duality principle for Gabor frames and the unitary extension principle for wavelet frames. This paper aims at establishing the dual Gramian analysis for frames in a general Hilbert space and subsequently characterizing the frame properties of a given system using the dual Gramian matrix generated by its elements. Consequently, many interesting results can be obtained for frames in Hilbert spaces, e.g., estimates of the frame bounds in terms of the frame elements and the duality principle. Moreover, this new characterization provides new insights into the unitary extension principle in a paper by Ron and Shen, e.g., the connection between the unitary extension principle and the duality principle in a weak sense. One application of such a connection is a simplification of the construction of multivariate tight wavelet frames from a given refinable mask. In contrast to the existing methods that require completing a unitary matrix with trigonometric polynomial entries from a given row, our method greatly simplifies the tight wavelet frame construction by converting it to a constant matrix completion problem. To illustrate its simplicity, the proposed construction scheme is used to construct a few examples of multivariate tight wavelet frames from box splines with certain desired properties, e.g., compact support, symmetry or anti-symmetry.


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Additional Information

Zhitao Fan
Affiliation: Department of Mathematics, National University of Singapore,10, Lower Kent Ridge Road, Singapore 119076, Singapore
Email: a0030819@nus.edu.sg

Hui Ji
Affiliation: Department of Mathematics, National University of Singapore,10, Lower Kent Ridge Road, Singapore 119076, Singapore
Email: matjh@nus.edu.sg

Zuowei Shen
Affiliation: Department of Mathematics, National University of Singapore,10, Lower Kent Ridge Road, Singapore 119076, Singapore
Email: matzuows@nus.edu.sg

DOI: https://doi.org/10.1090/mcom/2987
Received by editor(s): May 6, 2013
Received by editor(s) in revised form: April 26, 2014
Published electronically: June 23, 2015
Additional Notes: The work of the authors was partially supported by Singapore MOE Research Grant R-146-000-165-112 and R-146-000-154-112.
Article copyright: © Copyright 2015 American Mathematical Society