Solutions of a class of multiplicatively advanced differential equations. II: Fourier transforms (Q2111143)

Summary: 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. For Part I, see [\textit{D. W. Pravica} et al., C. R., Math., Acad. Sci. Paris 356, No. 7, 776--817 (2018; Zbl 1459.34146)].