Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms
Author
Pravica, David W.; Randriampiry, Njinasoa; Spurr, Michael J.
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,λðtÞ for μ and λ rational with λ > 0. These Wμ,λðtÞ 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,λðtÞ 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−4/3,1/3ðtÞ/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.
Date
2022-07-07
Citation:
APA:
Pravica, David W., & Randriampiry, Njinasoa, & Spurr, Michael J.. (July 2022).
Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms.
,
(),
-
. Retrieved from
http://hdl.handle.net/10342/10762
MLA:
Pravica, David W., and Randriampiry, Njinasoa, and Spurr, Michael J..
"Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms". .
. (),
July 2022.
June 29, 2024.
http://hdl.handle.net/10342/10762.
Chicago:
Pravica, David W. and Randriampiry, Njinasoa and Spurr, Michael J.,
"Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms," , no.
(July 2022),
http://hdl.handle.net/10342/10762 (accessed
June 29, 2024).
AMA:
Pravica, David W., Randriampiry, Njinasoa, Spurr, Michael J..
Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms. .
July 2022;
():
.
http://hdl.handle.net/10342/10762. Accessed
June 29, 2024.
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