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Generalized Matrix Spectral Factorization with Symmetry and Construction of Quasi-Tight Framelets over Algebraic Number Fields

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  • Ran Lu

    (Department of Mathematics, Hohai University, Nanjing 211100, China)

Abstract

The rational field Q is highly desired in many applications. Algorithms using the rational number field Q algebraic number fields use only integer arithmetics and are easy to implement. Therefore, studying and designing systems and expansions with coefficients in Q or algebraic number fields is particularly interesting. This paper discusses constructing quasi-tight framelets with symmetry over an algebraic field. Compared to tight framelets, quasi-tight framelets have very similar structures but much more flexibility in construction. Several recent papers have explored the structure of quasi-tight framelets. The construction of symmetric quasi-tight framelets directly applies the generalized spectral factorization of 2 × 2 matrices of Laurent polynomials with specific symmetry structures. We adequately formulate the latter problem and establish the necessary and sufficient conditions for such a factorization over a general subfield F of C , including algebraic number fields as particular cases. Our proofs of the main results are constructive and thus serve as a guideline for construction. We provide several examples to demonstrate our main results.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:6:p:919-:d:1360649
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    References listed on IDEAS

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    1. 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.
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