Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms
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...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2022-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2022/6721360 |
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| Summary: | 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 Wμ,λt for μ and λ rational with λ>0. These Wμ,λt have all moments vanishing and have a Fourier transform related to theta functions. For low parameter values derived from λ, the connection of the Wμ,λt 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−4/3,1/3t/W−4/3,1/30. A useful set of generalized q-Wallis formulas are developed that play a key role in this study of convergence. |
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| ISSN: | 1687-0409 |