In mathematics, a zonal polynomial is a multivariate symmetric homogeneous polynomial. The zonal polynomials form a basis of the space of symmetric polynomials. Zonal polynomials appear in special functions with matrix argument which on the other hand appear in matrixvariate distributions such as the Wishart distribution when integrating over compact Lie groups. The theory was started in multivariate statistics in the 1960s and 1970s in a series of papers by Alan Treleven James and his doctorial student Alan Graham Constantine.[1][2][3]
They appear as zonal spherical functions of the Gelfand pairs
(here, is the hyperoctahedral group) and , which means that they describe canonical basis of the double class
algebras and .
The zonal polynomials are the case of the C normalization of the Jack function.
^James, Alan Treleven (1961). "Zonal Polynomials of the Real Positive Definite Symmetric Matrices". Annals of Mathematics. 74: 456–469. doi:10.2307/1970291.
^James, Alan Treleven (1964). "Distributions of Matrix Variates and Latents Roots Derived from Normal Samples". Ann. Math. Statist. 35: 475–501. doi:10.1214/aoms/1177703550.
^Constantine, Alan Graham (1963). "Some Noncentral Distribution Problems in Multivariate Analysis". Ann. Math. Statist. 34: 1270–1285. doi:10.1214/aoms/1177703863.
zonalpolynomial is a multivariate symmetric homogeneous polynomial. The zonalpolynomials form a basis of the space of symmetric polynomials. Zonal polynomials...
plasma Zonalpolynomial, a symmetric multivariate polynomialZonal pelargonium, a type of pelargoniums Zonal tournaments in chess: see Interzonal#Zonal tournaments...
Jack polynomial, introduced by Henry Jack. The Jack polynomial is a homogeneous, symmetric polynomial which generalizes the Schur and zonalpolynomials, and...
\phi )=P_{\ell }(\cos \theta )} where Pℓ is a Legendre polynomial of degree ℓ. The general zonal spherical harmonic of degree ℓ is denoted by Z x ( ℓ )...
In mathematics, Gegenbauer polynomials or ultraspherical polynomials C(α) n(x) are orthogonal polynomials on the interval [−1,1] with respect to the weight...
the Macdonald polynomials become the (rescaled) zonal spherical functions for a semisimple p-adic group, or the Hall–Littlewood polynomials when the root...
harmonics Zonal spherical harmonics Multilinear polynomial Walsh, J. L. (1927). "On the Expansion of Harmonic Functions in Terms of Harmonic Polynomials". Proceedings...
radial dependence r ℓ {\displaystyle r^{\ell }} from the above-mentioned polynomial of degree ℓ {\displaystyle \ell } ; the remaining factor can be regarded...
of Legendre's differential equation. The Legendre polynomials and the associated Legendre polynomials are also solutions of the differential equation in...
In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K (often...
elementary functions, Bessel functions, and the classical orthogonal polynomials. A hypergeometric series is formally defined as a power series β 0 +...
reference ellipsoid and with z-axis in the direction of the polar axis. The zonal terms refer to terms of the form: P n 0 ( sin θ ) r n + 1 n = 0 , 1 ,...
concerns that the detection-line may be an artefact. The use of a 12th-order polynomial fit may have amplified noise and generated a false reading (see Runge's...
}^{m}} is the Legendre polynomial of degree l {\displaystyle l} with m = 0 {\displaystyle m=0} and is the associated Legendre polynomial with m > 0 {\displaystyle...
interconnection with the theory of differential equations. It is the special case for zonal spherical functions of the general Plancherel theorem for semisimple Lie...
for a simple RANS or LES case due to the RANS-LES switch. DES is a non-zonal approach and provides a single smooth velocity field across the RANS and...
that can perfectly reproduce wavefront modes in the form of Zernike polynomials. For predefined statistics of aberrations a deformable mirror with M...
meaning that any problem in that complexity class can be reduced in polynomial time to that problem. For example, generalized x × x chess has been proven...