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

Author:

Pravica David W.1ORCID,Randriampiry Njinasoa1ORCID,Spurr Michael J.1ORCID

Affiliation:

1. Department of Mathematics, East Carolina University, USA

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.

Funder

ECU Mathematics Department

Publisher

Hindawi Limited

Subject

Applied Mathematics,Analysis

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