Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms

ECU Author/Contributor (non-ECU co-authors, if there are any, appear on document)

David W. Pravica (Creator)

Njinasoa Randriampiry (Creator)

Michael J. Spurr (Creator)

Institution

East Carolina University (ECU )

Web Site: http://www.ecu.edu/lib/

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 Wµ,?ðtÞ for µ and ? rational with ? > 0. These Wµ,?ðtÞ have all moments vanishing and have a Fourier transform related to theta functions. For low parameter values derived from ?, the connection of the Wµ,?ðtÞ to the theory of wavelet frames is begun. For a second set of low parameter values derived from ?, 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-4/3,1/3ðtÞ/W-4/3,1/3ð0Þ. A useful set of generalized q-Wallis formulas are developed that play a key role in this study of convergence.

Additional Information

Publication

Other

Language: English

Date: 2023

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