Spherical functions
solid spherical harmonics, associated Legendre functions of the first and second kinds
Two linearly independent solutions and
of the differential equation
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where and
are complex constants, which arises in the solution of a class of partial differential equations by the method of separation of variables (cf. Separation of variables, method of). The points
are branching points of the solutions, in general. The spherical functions are particular cases of the hypergeometric functions (cf. Hypergeometric function):
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The spherical functions and
are defined and single-valued in the domains
and
, respectively, of the complex plane cut by the real axis from
to
.
If ,
,
, then the following functions are usually taken as solutions:
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where
are the values of the function
on the upper (lower) boundary of the cut.
When ,
are the Legendre polynomials. For zonal spherical functions see Spherical harmonics.
References
[1] | H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953) |
[2] | E. Jahnke, F. Emde, "Tables of functions with formulae and curves" , Dover, reprint (1945) (Translated from German) |
[3] | E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6 |
[4] | A. Krazer, W. Franz, "Transzendente Funktionen" , Akademie Verlag (1960) |
[5] | E.W. Hobson, "The theory of spherical and ellipsoidal harmonics" , Chelsea, reprint (1955) |
Comments
A more common usage of the phrase "spherical function" is as follows.
Let be a unimodular locally compact group and
a subgroup of
. Let
be an irreducible unitary representation of
on a Hilbert space
such that the
-fixed vectors in
form a one-dimensional subspace, spanned by a unit vector
. Then the
-bi-invariant function
on
defined by
is called a spherical function. Sometimes
is called a zonal spherical function, while the functions
(
) are also called spherical functions. Some authors call
an elementary spherical function, while all
-bi-invariant functions on
are called spherical functions.
The pair is a Gel'fand pair if, for all irreducible unitary representations of
, the subspace of
-fixed vectors in the representation space has dimension
or
. This is equivalent to the commutativity of the convolution algebra
of
-bi-invariant continuous functions on
with compact support. Now spherical functions are more generally defined as solutions
, not identically zero, of the functional equation
![]() | (*) |
where is the normalized Haar measure on
. These solutions include the spherical functions associated with irreducible unitary representations. Other solutions may be associated with irreducible non-unitary representations of
. The characters of the comutative algebra
are precisely the mappings
, where
is Haar measure on
and
is a solution of (*).
If is, moreover, a connected Lie group, then
is a Gel'fand pair if and only if the algebra
of
-invariant differential operators on the homogeneous space
is commutative. Then
is a solution of (*) if and only if it is
-bi-invariant,
,
, and the function
on
is a joint eigenfunction of the elements of
. In particular, if
is a connected real semi-simple LIe group and
is a maximal compact subgroup, then
is a Gel'fand pair,
is a Riemannian symmetric space, and much information is available about
and the sperical functions.
References
[a1] | J. Fauraut, "Analyse harmonique sur les paires de Gelfand et les espaces hyperboliques" , Anal. Harmonique , CIMPA (1982) pp. 315–446 |
[a2] | I.M. Gel'fand, "Spherical functions on symmetric spaces" Transl. Amer. Math. Soc. , 37 (1964) pp. 39–44 Dokl. Akad. Nauk SSSR , 70 (1950) pp. 5–8 |
[a3] | R. Godement, "Introduction aux traveaux de A. Selberg" Sem. Bourbaki , 144 (1957) |
[a4] | S. Helgason, "Groups and geometric analysis" , Acad. Press (1984) pp. Chapt. II, Sect. 4 |
Spherical functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spherical_functions&oldid=48775