In the mathematical study of rotational symmetry, the zonal spherical harmonics are special spherical harmonics that are invariant under the rotation through a particular fixed axis. The zonal spherical functions are a broad extension of the notion of zonal spherical harmonics to allow for a more general symmetry group.
On the two-dimensional sphere, the unique zonal spherical harmonic of degree ℓ invariant under rotations fixing the north pole is represented in spherical coordinates by
where Pℓ is a Legendre polynomial of degree ℓ. The general zonal spherical harmonic of degree ℓ is denoted by , where x is a point on the sphere representing the fixed axis, and y is the variable of the function. This can be obtained by rotation of the basic zonal harmonic
In n-dimensional Euclidean space, zonal spherical harmonics are defined as follows. Let x be a point on the (n−1)-sphere. Define to be the dual representation of the linear functional
in the finite-dimensional Hilbert space Hℓ of spherical harmonics of degree ℓ. In other words, the following reproducing property holds:
for all Y ∈ Hℓ. The integral is taken with respect to the invariant probability measure.
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zonalsphericalharmonics are special sphericalharmonics that are invariant under the rotation through a particular fixed axis. The zonalspherical functions...
scientific fields. The table of sphericalharmonics contains a list of common sphericalharmonics. Since the sphericalharmonics form a complete set of orthogonal...
In mathematics, a zonalspherical function or often just spherical function is a function on a locally compact group G with compact subgroup K (often a...
\mathbf {y} )} are sphericalharmonics, when regarded as a function of x only. They are, in fact, exactly the zonalsphericalharmonics, up to a normalizing...
Spherical function can refer to SphericalharmonicsZonalspherical function This disambiguation page lists mathematics articles associated with the same...
portal Harmonic function SphericalharmonicsZonalsphericalharmonics Multilinear polynomial Walsh, J. L. (1927). "On the Expansion of Harmonic Functions...
made by modeling the entire surface as an oblate spheroid, using sphericalharmonics to approximate the geoid, or modeling a region with a best-fit reference...
image, the higher average elevation of the southern hemisphere (zonalsphericalharmonic degree 1) has been removed to highlight the elevation contrast...
classical harmonic analysis and has a similarly close interconnection with the theory of differential equations. It is the special case for zonalspherical functions...
Kirillov orbit theory Discrete series representation Zonalspherical function "Noncommutative harmonic analysis: in honor of Jacques Carmona", Jacques Carmona...
Pmn for 1 ≤ m ≤ n are the associated Legendre functions. The first sphericalharmonics with n = 0, 1, 2, 3 are presented in the table below. [Note that...
is the Rossby parameter, k is the zonal wavenumber, and ℓ is the meridional wavenumber. It is noted that the zonal phase speed of Rossby waves is always...
investigations in the USSR and Japan. Work on the abstract theory of spherical functions published in 1952 proved very influential in subsequent work...
theory of SL2(R) Representations of the Lorentz group Stone–von Neumann theorem Unitary representation of a star Lie superalgebra Zonalspherical function...
the years models consisting of an increasing but limited number of sphericalharmonics have been produced. Maps produced have included free-air gravity...
equilibrium on an uninfluenced spherical bowl and made the addition of finding the potential at any point of space in zonalharmonics. He died at the College...
atmosphere occurs in an approximate square wave profile and so is rich in harmonics. When this pattern is decomposed into separate frequency components using...
of it from spherical symmetry. From the point of view of satellite dynamics, of particular relevance are the so-called even zonalharmonic coefficients...
directions. Wave trapping is the result of the Earth's rotation and its spherical shape which combine to cause the magnitude of the Coriolis force to increase...
coset space of S L ( 2 , R ) {\displaystyle SL(2,\mathbb {R} )} (see Zonalspherical function). Actually the Fourier transform on L 1 ( G / / K ) {\displaystyle...
is expressible in terms of elliptic functions (1915) A formula in zonalharmonics (1916–17) Potts, R. B. "Wilton, John Raymond (1884–1944)". Australian...
polynomials as a special case, and when considered in the form of sphericalharmonics, these polynomials reflect, in a certain sense, the symmetry properties...
and the Hough functions degenerate to sphericalharmonics; e.g., mode (1, -2) develops to the sphericalharmonic P11(θ), mode (2, 2) becomes P22(θ), with...
Earth's gravitational potential to a few zonalharmonic terms. The atmosphere is usually modeled as a static, spherical density field that exponentially decays...
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