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Generalized Matrix Spectral Factorization with Symmetry and Construction of Quasi-Tight Framelets over Algebraic Number Fields.
- Source :
- Mathematics (2227-7390); Mar2024, Vol. 12 Issue 6, p919, 29p
- Publication Year :
- 2024
-
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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 22277390
- Volume :
- 12
- Issue :
- 6
- Database :
- Complementary Index
- Journal :
- Mathematics (2227-7390)
- Publication Type :
- Academic Journal
- Accession number :
- 176368724
- Full Text :
- https://doi.org/10.3390/math12060919