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 de...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/890456 |
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| author | David W. Pravica Njinasoa Randriampiry Michael J. Spurr |
| author_facet | David W. Pravica Njinasoa Randriampiry Michael J. Spurr |
| author_sort | David W. Pravica |
| collection | DOAJ |
| description | 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≤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. |
| format | Article |
| id | doaj-art-dc9de163380d4f62ac3c47b094a30c4a |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-dc9de163380d4f62ac3c47b094a30c4a2025-02-03T06:00:42ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/890456890456Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta FunctionDavid W. Pravica0Njinasoa Randriampiry1Michael J. Spurr2Mathematics Department, East Carolina University, Greenville, NC 27858, USAMathematics Department, East Carolina University, Greenville, NC 27858, USAMathematics Department, East Carolina University, Greenville, NC 27858, USAThe 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≤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.http://dx.doi.org/10.1155/2014/890456 |
| spellingShingle | David W. Pravica Njinasoa Randriampiry Michael J. Spurr Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function Abstract and Applied Analysis |
| title | Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function |
| title_full | Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function |
| title_fullStr | Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function |
| title_full_unstemmed | Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function |
| title_short | Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function |
| title_sort | smooth wavelet approximations of truncated legendre polynomials via the jacobi theta function |
| url | http://dx.doi.org/10.1155/2014/890456 |
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