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Compactly supported Parseval framelets with symmetry associated to Ed(2)(Z) matrices

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  • San Antolín, A.
  • Zalik, R.A.

Abstract

Let d ≥ 1. For any A∈Zd×d such that |detA|=2, we construct two families of Parseval wavelet frames with two generators. These generators have compact support, any desired number of vanishing moments, and any given degree of regularity. The first family is real valued while the second family is complex valued. To construct these families we use Daubechies low pass filters to obtain refinable functions, and adapt methods employed by Chui and He and Petukhov for dyadic dilations to this more general case. We also construct several families of Parseval wavelet frames with three generators having various symmetry properties. Our constructions are based on the same refinable functions and on techniques developed by Han and Mo and by Dong and Shen for the univariate case with dyadic dilations.

Suggested Citation

  • San Antolín, A. & Zalik, R.A., 2018. "Compactly supported Parseval framelets with symmetry associated to Ed(2)(Z) matrices," Applied Mathematics and Computation, Elsevier, vol. 325(C), pages 179-190.
    • Handle: RePEc:eee:apmaco:v:325:y:2018:i:c:p:179-190
    DOI: 10.1016/j.amc.2017.12.008
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    References listed on IDEAS

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    1. Skopina, M., 2017. "On construction of multivariate Parseval wavelet frames," Applied Mathematics and Computation, Elsevier, vol. 301(C), pages 1-11.
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    Cited by:

    1. Krivoshein, A.V., 2019. "From frame-like wavelets to wavelet frames keeping approximation properties and symmetry," Applied Mathematics and Computation, Elsevier, vol. 344, pages 204-218.
    2. Zhihua Zhang, 2021. "Splitting of Framelets and Framelet Packets," Mathematics, MDPI, vol. 9(7), pages 1-10, March.
    3. Zhihua Zhang, 2023. "The Improvement of the Discrete Wavelet Transform," Mathematics, MDPI, vol. 11(8), pages 1-12, April.
    4. Zhihua Zhang, 2022. "Non-Separable Meyer-like Wavelet Frames," Mathematics, MDPI, vol. 10(13), pages 1-14, June.
    5. Ran Lu, 2024. "Generalized Matrix Spectral Factorization with Symmetry and Construction of Quasi-Tight Framelets over Algebraic Number Fields," Mathematics, MDPI, vol. 12(6), pages 1-29, March.
    6. Zhihua Zhang, 2021. "Framelet Sets and Associated Scaling Sets," Mathematics, MDPI, vol. 9(21), pages 1-10, November.
    7. Zhihua Zhang, 2021. "Characterization of Frequency Domains of Bandlimited Frame Multiresolution Analysis," Mathematics, MDPI, vol. 9(9), pages 1-9, May.

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