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

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  • David W. Pravica
  • Njinasoa Randriampiry
  • Michael J. Spurr

Abstract

The family of n th 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 n th degree Legendre polynomials. The n th order q -Legendre polynomials are shown to have vanishing k th moments for , as does the n th 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.

Suggested Citation

  • David W. Pravica & Njinasoa Randriampiry & Michael J. Spurr, 2014. "Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-24, October.
    • Handle: RePEc:hin:jnlaaa:890456
    DOI: 10.1155/2014/890456
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