# Prolate Spheroidal Wave Functions

In mathematics, the prolate spheroidal wave functions are a set of functions derived by timelimiting and lowpassing, and a second timelimit operation. Let denote the time truncation operator, such that iff x is timelimited within . Similarly, let denote an ideal low-pass filtering operator, such that iff x is bandlimited within . The operator turns out to be linear, bounded and self-adjoint. For we denote with the n-th eigenfunction, defined as

where are the associated eigenvalues. The timelimited functions are the Prolate Spheroidal Wave Functions (PSWFs).

These functions are also encountered in a different context. When solving the Helmholtz equation, by the method of separation of variables in prolate spheroidal coordinates, with:

and .

the solution can be written as the product of a radial spheroidal wavefunction and an angular spheroidal wavefunction by with .

The radial wavefunction satisfies the linear ordinary differential equation:

The eigenvalue of this Sturm-Liouville differential equation is fixed by the requirement that must be finite for .

The angular wavefunction satisfies the differential equation:

It is the same differential equation as in the case of the radial wavefunction. However, the range of the variable is different (in the radial wavefunction, ) in the angular wavefunction ).

For these two differential equations reduce to the equations satisfied by the associated Legendre polynomials. For, the angular spheroidal wavefunctions can be expanded as a series of Legendre functions.

Let us note that if one writes, the function satisfies the following linear ordinary differential equation:

which is known as the spheroidal wave equation. This auxiliary equation is used for instance by Stratton in his 1935 article.

There are different normalization schemes for spheroidal functions. A table of the different schemes can be found in Abramowitz and Stegun p. 758. Abramowitz and Stegun (and the present article) follow the notation of Flammer.

In the case of oblate spheroidal coordinates the solution of the Helmholtz equation yields oblate spheroidal wavefunctions.

Originally, the spheroidal wave functions were introduced by C. Niven in 1880 when studying the conduction of heat in an ellipsoid of revolution, which lead to a Helmholtz equation in spheroidal coordinates.

Prolate spheroidal wave functions whose domain is a (portion of) the surface of the unit sphere are more generally called "Slepian functions" (see also Spectral concentration problem). These are of great utility in disciplines such as geodesy or cosmology.

### Other articles related to "prolate spheroidal, spheroidal, spheroidal wave functions, prolate spheroidal wave functions":

Prolate Spheroidal Coordinates
... Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating a spheroid around its major axis, i.e ... Rotation about the other axis produces the oblate spheroidal coordinates ... Prolate spheroidal coordinates can be used to solve various partial differential equations in which the boundary conditions match its symmetry and shape, such as solving for a field produced ...
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... Spheroidal wave functions are solutions of the Helmholtz equation that are found by writing the equation in spheroidal coordinates and applying the technique of ... They are called oblate spheroidal wave functions if oblate spheroidal coordinates are used and prolate spheroidal wave functions if prolate spheroidal coordinates are used ... If instead of the Helmholtz equation, the Laplace equation is solved in spheroidal coordinates using the method of separation of variables, the ...

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