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Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms.
- Source :
- Abstract & Applied Analysis; 7/7/2022, p1-49, 49p
- Publication Year :
- 2022
-
Abstract
- For a wide class of solutions to multiplicatively advanced differential equations (MADEs), a comprehensive set of relations is established between their Fourier transforms and Jacobi theta functions. In demonstrating this set of relations, the current study forges a systematic connection between the theory of MADEs and that of special functions. In a large subset of the general case, we introduce a new family of Schwartz wavelet MADE solutions W μ , λ t for μ and λ rational with λ > 0. These W μ , λ t have all moments vanishing and have a Fourier transform related to theta functions. For low parameter values derived from λ , the connection of the W μ , λ t to the theory of wavelet frames is begun. For a second set of low parameter values derived from λ , the notion of a canonical extension is introduced. A number of examples are discussed. The study of convergence of the MADE solution to the solution of its analogous ODE is begun via an in depth analysis of a normalized example W − 4 / 3 , 1 / 3 t / W − 4 / 3 , 1 / 3 0 . A useful set of generalized q -Wallis formulas are developed that play a key role in this study of convergence. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 10853375
- Database :
- Complementary Index
- Journal :
- Abstract & Applied Analysis
- Publication Type :
- Academic Journal
- Accession number :
- 157864648
- Full Text :
- https://doi.org/10.1155/2022/6721360