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 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.