Constructions of Frames by Extension Principles

The extension principles make constructions of wavelet frame systems simple.  All wavelet frames used in our applications are designed by the unitary extension principle. We constructed wavelet frames from splines, box splines, and pseudo-splines in [3,5,6,11,12], sheaelets in [7],  and wavelet frames in Sobolev spaces in [8,9]. We also developed wavelet frames on the graphs in [1],    Furthermore, by applying the unitary extension principle, we discovered in [2,4] that most often seen digital Gabor filters (e.g. local discrete Fourier transform and discrete Cosine transform) do generate MRA-based wavelet tight frames in square-integrable function space.  The corresponding refinable functions and wavelets can be explicitly given. Discrete tight frames generated by such filters with both wavelet and Gabor structures were proven efficient in image processing and recovery. (see e.g. [2,4].)

    1. Hui Ji, Zuowei Shen, Yufei Zhao, Multi-scale discrete framelet transform for graph-structured signals,  Multiscale Modeling, and Simulation: A SIAM Interdisciplinary Journal, 18(3) (2020), 1210-1241. PDF
    2. Hui Ji, Zuowei Shen,Yufei Zhao, Digital Gabor Filters with MRA structure, Multiscale Modeling and Simulation: A SIAM Interdisciplinary 16(1), (2018), 452-476. PDF
    3. Bin Han, Qingtang Jiang, Zuowei Shen, Xiaosheng Zhuan, Symmetric canonical quincunx tight framelets with high vanishing moments and smoothness, Mathematics of Computation, 87 (2018) 347-379.PDF
    4. Hui Ji, Zuowei Shen, Yufei Zhao, Directional Frames for Image Recovery: Multi-scale Discrete Gabor Frames, Journal of Fourier Analysis and Applications, 23(4), (2017), 729-757. PDF
    5. Qingtang Jiang, Zuowei Shen, Tight wavelet frames in low dimensions with canonical filters, Journal of Approximation Theory 196 (2015) 55-78. PDF
    6. Zuowei Shen, Zhiqiang Xu, On B-spline framelets derived from the unitary extension principle, SIAM Journal on Mathematical Analysis, 45(11) (2013), 127-151. PDF
    7. Bin Han, Gitta Kutyniko, Zuowei Shen, Adaptive multiresolution analysis structures and shearlet systems, SIAM Journal on Numerical Analysis, 49 (5), (2011), 1921-1946.PDF
    8. Bin Han, Zuowei Shen, Dual wavelet frames and Riesz bases in Sobolev spaces, Constructive Approximation, 29(3) (2009), 369-406. PDF
    9. Bin Han, Zuowei Shen, Characterization of Sobolev spaces of arbitrary smoothness using nonstationary tight wavelet frames, Israel Journal of Mathematics, 172 (2009), 371-398. PDF
    10. Bin Han, Zuowei Shen, Compactly supported symmetric C^\infty wavelets with spectral approximation order, SIAM Journal on Mathematical Analysis, 40(3) (2008), 905-938. PDF
    11. Bin Dong, Zuowei Shen, Pseudo-splines, wavelets and framelets, Applied and Computational Harmonic Analysis, 22 (1) (2007), 78-104. pseudospline1.pdf
    12. Amos Ron, Zuowei Shen, Compactly supported tight affine spline frames in L2(Rd), Mathematics of Computation, 67 (1998), 191-207. PDF

The constructions of non-redundant multiresolution analysis-based wavelet systems, such as pre-wavelets, orthogonal and biorthogonal wavelets, can be founded here.