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Abstract and Applied Analysis
Volume 2014, Article ID 890456, 24 pages
Research Article

Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function

Mathematics Department, East Carolina University, Greenville, NC 27858, USA

Received 7 March 2014; Accepted 14 July 2014; Published 16 October 2014

Academic Editor: Cristina Pignotti

Copyright © 2014 David W. Pravica et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 , 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.