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

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.ECU Libraries Open Access Publishing Support Fun

Smooth wavelet approximations of truncated Legendre polynomials via the Jacobi theta function

The family of nth order q-Legendre polynomials are introduced. They are shown to be obtainable from the Jacobi theta function and to satisfy recursion relations and multiplicatively advanced differential equations (MADEs) that are analogues of the recursion relations and ODEs satisfied by the nth degree Legendre polynomials. The nth order q-Legendre polynomials are shown to have vanishing kth moments for 0...4;k<n , as does the nth degree truncated Legendre polynomial. Convergence results are obtained, approximations are given, a reciprocal symmetry is shown, and nearly orthonormal frames are constructed. Conditions are given under which a MADE remains a MADE under inverse Fourier transform. This is used to construct new wavelets as solutions of MADEs

Strategies for the hyperpolarization of acetonitrile and related Ligands by SABRE

(Chemical Equation Presented) We report on a strategy for using SABRE (signal amplification by reversible exchange) for polarizing 1H and 13C nuclei of weakly interacting ligands which possess biologically relevant and nonaromatic motifs. We first demonstrate this via the polarization of acetonitrile, using Ir(IMes)(COD)Cl as the catalyst precursor, and confirm that the route to hyperpolarization transfer is via the J-coupling network. We extend this work to the polarization of propionitrile, benzylnitrile, benzonitrile, and trans-3-hexenedinitrile in order to assess its generality. In the 1H NMR spectrum, the signal for acetonitrile is enhanced 8-fold over its thermal counterpart when [Ir(H)2(IMes)(MeCN)3]+ is the catalyst. Upon addition of pyridine or pyridine-d5, the active catalyst changes to [Ir(H)2(IMes)- (py)2(MeCN)]+ and the resulting acetonitrile 1H signal enhancement increases to 20- and 60-fold, respectively. In 13C NMR studies, polarization transfers optimally to the quaternary 13C nucleus of MeCN while the methyl 13C is hardly polarized. Transfer to 13C is shown to occur first via the 1H - 1H coupling between the hydrides and the methyl protons and then via either the 2J or 1J couplings to the respective 13Cs, of which the 2J route is more efficient. These experimental results are rationalized through a theoretical treatment which shows excellent agreement with experiment. In the case of MeCN, longitudinal two-spin orders between pairs of 1H nuclei in the three-spin methyl group are created. Two-spin order states, between the 1H and 13C nuclei, are also created, and their existence is confirmed for Me13CN in both the 1H and 13C NMR spectra using the Only Parahydrogen Spectroscopy protocol

Eigenfunction Families and Solution Bounds for Multiplicatively Advanced Differential Equations

A family of Schwartz functions W ( t ) are interpreted as eigensolutions of MADEs in the sense that W ( &delta; ) ( t ) = E W ( q &gamma; t ) where the eigenvalue E &isin; R is independent of the advancing parameter q &gt; 1 . The parameters &delta; , &gamma; &isin; N are characteristics of the MADE. Some issues, which are related to corresponding q-advanced PDEs, are also explored. In the limit that q &rarr; 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 &isin; 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

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

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,?ðtÞ for µ and ? rational with ? &gt; 0. These Wµ,?ðtÞ have all moments vanishing and have a Fourier transform related to theta functions. For low parameter values derived from ?, the connection of the Wµ,?ðtÞ to the theory of wavelet frames is begun. For a second set of low parameter values derived from ?, 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

Smooth wavelet approximations of truncated Legendre polynomials via the Jacobi theta function

The family of nth order q-Legendre polynomials are introduced. They are shown to be obtainable from the Jacobi theta function and to satisfy recursion relations and multiplicatively advanced differential equations (MADEs) that are analogues of the recursion relations and ODEs satisfied by the nth degree Legendre polynomials. The nth order q-Legendre polynomials are shown to have vanishing kth moments for 0...4;k&lt;n , as does the nth degree truncated Legendre polynomial. Convergence results are obtained, approximations are given, a reciprocal symmetry is shown, and nearly orthonormal frames are constructed. Conditions are given under which a MADE remains a MADE under inverse Fourier transform. This is used to construct new wavelets as solutions of MADEs