orthogonal polynomials on the unit circle
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21—25 of 25 matching pages
21: 10.60 Sums
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►Then with again denoting the Legendre polynomial of degree ,
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10.60.1
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10.60.2
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10.60.3
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10.60.7
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22: 14.30 Spherical and Spheroidal Harmonics
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and () are known as oblate spheroidal harmonics of the first and second kinds, respectively.
Segura and Gil (1999) introduced the scaled oblate spheroidal harmonics and which are real when and .
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Orthogonality
… ►In general, spherical harmonics are defined as the class of homogeneous harmonic polynomials. See Andrews et al. (1999, Chapter 9). …23: 3.11 Approximation Techniques
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§3.11(i) Minimax Polynomial Approximations
… ►They enjoy an orthogonal property with respect to integrals: …as well as an orthogonal property with respect to sums, as follows. When and , , … ► …24: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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, corresponding to distinct eigenvalues, are orthogonal: i.
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►, of unit multiplicity, unless otherwise stated.
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►Orthogonality and normalization may then be chosen such that analogous to (1.18.19) and (1.18.20), we have
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►Then orthogonality and normalization relations are
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►Note that eigenfunctions for distinct (necessarily real) eigenvalues of a self-adjoint operator are mutually orthogonal.
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25: 28.31 Equations of Whittaker–Hill and Ince
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