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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. [ABSTRACT FROM AUTHOR]

A family of Schwartz functions W (t) are interpreted as eigensolutions of MADEs in the sense that W (δ) (t) = E W (q γ t) where the eigenvalue E ∈ R is independent of the advancing parameter q > 1 . The parameters δ , γ ∈ N are characteristics of the MADE. Some issues, which are related to corresponding q-advanced PDEs, are also explored. In the limit that q → 1 + we show convergence of MADE eigenfunctions to solutions of ODEs, which involve only simple exponentials and trigonometric functions. The limit eigenfunctions ( q = 1 + ) are not Schwartz, thus convergence is only uniform in t ∈ R on compact sets. An asymptotic analysis is provided for MADEs which indicates how to extend solutions in a neighborhood of the origin t = 0 . Finally, an expanded table of Fourier transforms is provided that includes Schwartz solutions to MADEs. [ABSTRACT FROM AUTHOR]

The multiplicatively advanced differential equations (MADEs) of form f ( n ) ( t ) = α f ( β t ) with α ≠ 0 , β > 1 are studied along with a class of their solutions of type f μ , λ ( t ) defined on [ 0 , ∞ ) . For λ ∈ Q + , μ ∈ R , the solutions f μ , λ ( t ) are extended to ( − ∞ , ∞ ) in a non-unique manner to obtain Schwartz wavelet solutions F μ , λ ( t ) of the original MADE, with all moments of F μ , λ ( t ) vanishing. Examples are studied in detail. The Fourier transform of each F μ , λ ( t ) is computed and, in a number of examples, is related to the Jacobi theta function. Additional conditions sufficient for the uniqueness of certain MADE initial value problems are given. Conditions for decay and non-decay at −∞ are obtained. Decay rates at ±∞ in terms of familiar functions are established. [ABSTRACT FROM AUTHOR]