## Abstract and Applied Analysis

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

#### Abstract

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\le k, 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 890456, 24 pages.

Dates
First available in Project Euclid: 27 February 2015

https://projecteuclid.org/euclid.aaa/1425049836

Digital Object Identifier
doi:10.1155/2014/890456

Mathematical Reviews number (MathSciNet)
MR3272222

Zentralblatt MATH identifier
07023246

#### Citation

Pravica, David W.; Randriampiry, Njinasoa; Spurr, Michael J. Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function. Abstr. Appl. Anal. 2014 (2014), Article ID 890456, 24 pages. doi:10.1155/2014/890456. https://projecteuclid.org/euclid.aaa/1425049836