The prolate spheroidal wave functions are
eigenfunctions
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
of the Laplacian in prolate spheroidal coordinates, adapted to boundary conditions on certain ellipsoids of revolution (an ellipse rotated around its long axis, “cigar shape“). Related are the
oblate spheroidal wave functions In applied mathematics, oblate spheroidal wave functions (like also prolate spheroidal wave functions and other related functions) are involved in the solution of the Helmholtz equation in oblate spheroidal coordinates. When solving this equation, ...
(“pancake shaped” ellipsoid).
Solutions to the wave equation
Solve the
Helmholtz equation,
, by the method of separation of variables in
prolate spheroidal coordinates
Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are loc ...
,
, with:
:
:
:
and
,
, and
. Here,
is the interfocal distance of the elliptical cross section of the prolate spheroid.
Setting
, the solution
can be written
as the product of
, a radial spheroidal wave function
and an angular spheroidal wave function
.
The radial wave function
satisfies the linear
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
:
:
The angular wave function satisfies the differential equation:
:
It is the same differential equation as in the case of the radial wave function. However, the range of the variable is different: in the radial wave function,
, while in the angular wave function,
. The eigenvalue
of this
Sturm–Liouville problem is fixed by the requirement that
must be finite for
.
For
both differential equations reduce to the equations satisfied by the
associated Legendre polynomials. For
, the angular spheroidal wave functions can be expanded as a series of Legendre functions.
If one writes
, the function
satisfies
:
which is known as the
spheroidal wave equation. This auxiliary equation has been used by Stratton.
Band-limited signals
In signal processing, the prolate spheroidal wave functions (PSWF) are useful as eigenfunctions of a time-limiting operation followed by a low-pass filter. Let
denote the time truncation operator, such that
if and only if
has support on