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

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    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.
    URI
    http://hdl.handle.net/10342/10762
    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

    Display/Hide MLA, Chicago and APA citation formats.

    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. February 02, 2023. 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 February 02, 2023).
    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 February 02, 2023.
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