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

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 ${\mathcal{W}}_{\mu ,\lambda }\left(t\right)$ for $\mu$ and $\lambda$ rational with $\lambda >0$ . These ${\mathcal{W}}_{\mu ,\lambda }\left(t\right)$ have all moments vanishing and have a Fourier transform related to theta functions. For low parameter values derived from $\lambda$ , the connection of the ${\mathcal{W}}_{\mu ,\lambda }\left(t\right)$ to the theory of wavelet frames is begun. For a second set of low parameter values derived from $\lambda$ , 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 ${\mathcal{W}}_{-4/3,1/3}\left(t\right)/{\mathcal{W}}_{-4/3,1/3}\left(0\right)$ . A useful set of generalized $q$ -Wallis formulas are developed that play a key role in this study of convergence.