In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

study of

rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...

, the zonal spherical harmonics are special

spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a ...

that are invariant under the rotation through a particular fixed axis. The

zonal spherical function 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 a maximal compact subgroup) that arises as the matrix coefficient of a ''K''-invariant vect ...

s are a broad extension of the notion of zonal spherical harmonics to allow for a more general

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient ...

. 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 In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...

Z^(\theta,\phi) = P_\ell(\cos\theta)

where is a

Legendre polynomial In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ...

of degree . The general zonal spherical harmonic of degree ℓ is denoted by

Z^_(\mathbf)

, 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

Z^(\theta,\phi).

In ''n''-dimensional Euclidean space, zonal spherical harmonics are defined as follows. Let x be a point on the (''n''−1)-sphere. Define

Z^_

to be the

dual representation In mathematics, if is a group and is a linear representation of it on the vector space , then the dual representation is defined over the dual vector space as follows: : is the transpose of , that is, = for all . The dual representation ...

of the linear functional

P\mapsto P(\mathbf)

in the finite-dimensional

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

H_ℓ of spherical harmonics of degree ℓ. In other words, the following reproducing property holds:

Y(\mathbf) = \int_ Z^_(\mathbf)Y(\mathbf)\,d\Omega(y)

for all . The integral is taken with respect to the invariant probability measure.

Relationship with harmonic potentials

The zonal harmonics appear naturally as coefficients of the

Poisson kernel In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the deriva ...

for the unit ball in R^''n'': for x and y unit vectors,

\frac\frac = \sum_^\infty r^k Z^_(\mathbf),

where

\omega_

is the surface area of the (n-1)-dimensional sphere. They are also related to the

Newton kernel In mathematics, the Newtonian potential or Newton potential is an operator in vector calculus that acts as the inverse to the negative Laplacian, on functions that are smooth and decay rapidly enough at infinity. As such, it is a fundamental obje ...

via

\frac = \sum_^\infty c_ \fracZ_^(\mathbf/, \mathbf, )

where and the constants are given by

c_ = \frac\frac.

The coefficients of the Taylor series of the Newton kernel (with suitable normalization) are precisely the

ultraspherical polynomials In mathematics, Gegenbauer polynomials or ultraspherical polynomials ''C''(''x'') are orthogonal polynomials on the interval minus;1,1with respect to the weight function (1 − ''x''2)''α''–1/2. They generalize Legendre polynomi ...

. Thus, the zonal spherical harmonics can be expressed as follows. If , then

Z^_(\mathbf) = \fracC_\ell^(\mathbf\cdot\mathbf)

where are the constants above and

C_\ell^

is the ultraspherical polynomial of degree ℓ.

Properties

*The zonal spherical harmonics are rotationally invariant, meaning that

Z^_(R\mathbf) = Z^_(\mathbf)

for every orthogonal transformation ''R''. Conversely, any function on that is a spherical harmonic in ''y'' for each fixed ''x'', and that satisfies this invariance property, is a constant multiple of the degree zonal harmonic. *If ''Y''₁, ..., ''Y''_''d'' is an

orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...

of , then

Z^_(\mathbf) = \sum_^d Y_k(\mathbf)\overline.

*Evaluating at gives

Z^_(\mathbf) = \omega_^ \dim \mathbf_\ell.

References

* {{citation, last1=Stein, first1=Elias, authorlink1=Elias Stein, first2=Guido, last2=Weiss, authorlink2=Guido Weiss, title=Introduction to Fourier Analysis on Euclidean Spaces, publisher=Princeton University Press, year=1971, isbn=978-0-691-08078-9, location=Princeton, N.J., url-access=registration, url=https://archive.org/details/introductiontofo0000stei. Rotational symmetry Special hypergeometric functions