zonal spherical harmonics

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1: 14.30 Spherical and Spheroidal Harmonics

§14.30 Spherical and Spheroidal Harmonics

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§14.30(i) Definitions

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§14.30(iii) Sums

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2: 35.4 Partitions and Zonal Polynomials

§35.4 Partitions and Zonal Polynomials

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Normalization

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Orthogonal Invariance

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Summation

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Mean-Value

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3: 37.11 Spherical Harmonics

§37.11 Spherical Harmonics

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Zonal Functions

… ►A spherical harmonic $Y\in {\mathscr{H}}_n^{0,d}$ is zonal iff … ►

Jacobi Polynomials as Spherical Harmonics

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Fourier Transform Involving Spherical Harmonics

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4: 18.38 Mathematical Applications

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Zonal Spherical Harmonics

►Ultraspherical polynomials are zonal spherical harmonics. …

5: 35.11 Tables

§35.11 Tables

►Tables of zonal polynomials are given in James (1964) for $|\kappa |\le 6$, Parkhurst and James (1974) for $|\kappa |\le 12$, and Muirhead (1982, p. 238) for $|\kappa |\le 5$. Each table expresses the zonal polynomials as linear combinations of monomial symmetric functions.

6: 35.12 Software

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Demmel and Koev (2006). Computation of zonal polynomials in MATLAB.

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Stembridge (1995). Maple software for zonal polynomials.

►For an algorithm to evaluate zonal polynomials, and an implementation of the algorithm in Maple by Zeilberger, see Lapointe and Vinet (1996).

7: Bibliography T

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Zonal Polynomials. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 4, Institute of Mathematical Statistics, Hayward, CA.

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External Links:

ISBN 0-940600-05-6, FullText, MathReview (A. W. Davis), ZentralBlatt

Referenced by:

§35.4(i), §35.9.

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LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.

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Notes:

Single-precision Fortran.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

9th item.

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Harmonic Analysis on Symmetric Spaces and Applications. II. Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-96663-3, MathReview (Stephen Gelbart), ZentralBlatt

Referenced by:

§35.1, §35.5(ii), Notations K.

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BibTeX

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Hyperspherical elliptic harmonics and their relation to the Heun equation. Phys. Rev. A 63 (032510), pp. 1–8.

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External Links:

Document

Referenced by:

§31.17(ii).

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Representation Theory and Harmonic Analysis. Contemporary Mathematics, Vol. 191, American Mathematical Society, Providence, RI.

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Notes:

Papers from the conference held in honor of Ray A. Kunze at the AMS Special Session, Cincinnati, Ohio, January 12–14, 1994

External Links:

ISBN 0-8218-0310-7, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

Profile Donald St. P. Richards.

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BibTeX

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8: 15.17 Mathematical Applications

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§15.17(iii) Group Representations

►For harmonic analysis it is more natural to represent hypergeometric functions as a Jacobi function (§15.9(ii)). …First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL$(2,ℝ)$, and spherical functions on certain nonsymmetric Gelfand pairs. Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. …

9: 35.10 Methods of Computation

… ►For small values of $\Vert 𝐓\Vert$ the zonal polynomial expansion given by (35.8.1) can be summed numerically. … ►Koev and Edelman (2006) utilizes combinatorial identities for the zonal polynomials to develop computational algorithms for approximating the series expansion (35.8.1). …

10: Bibliography G

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A harmonic mean inequality for the gamma function. SIAM J. Math. Anal. 5 (2), pp. 278–281.

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External Links:

Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§5.6(i).

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BibTeX

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A code to evaluate prolate and oblate spheroidal harmonics. Comput. Phys. Comm. 108 (2-3), pp. 267–278.

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Notes:

Double-precision Fortran, maximum accuracy 18S

External Links:

Document, Code, ZentralBlatt

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5th item, 1st item.

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Evaluation of toroidal harmonics. Comput. Phys. Comm. 124 (1), pp. 104–122.

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External Links:

Document, Code, ZentralBlatt

Referenced by:

4th item.

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DTORH3 2.0: A new version of a computer program for the evaluation of toroidal harmonics. Comput. Phys. Comm. 139 (2), pp. 186–191.

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External Links:

Document, Code, ZentralBlatt

Referenced by:

2nd item.

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Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions. Trans. Amer. Math. Soc. 301 (2), pp. 781–811.

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External Links:

ISSN 0002-9947, FullText, MathReview (Wayne J. Holman, III), ZentralBlatt

Referenced by:

§35.8(i), §35.8(ii), §35.8(iv), Ch.35.

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