Prolate spheroidal wave functions

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The prolate spheroidal wave functions are a set of functions derived by timelimiting and lowpassing, and a second timelimit operation. Let $Q T$ denote the time truncation operator, such that $x = Q T x$ iff x is timelimited within $[ - T / 2; T / 2]$ . Similarly, let $P W$ denote an ideal low-pass filtering operator, such that $x = P W x$ iff x is bandlimited within $[ - W; W]$ . The operator $Q T P W Q T$ turns out to be linear, bounded and self-adjoint. For $n=1,2,\ldots$ we denote with $ψ n$ the n-th eigenfunction, defined as

$\ Q_T P_W Q_T \psi_n=\lambda_n\psi_n,$

where ${λ n} n$ are the associated eigenvalues. The timelimited functions ${ψ n} n$ are the Prolate Spheroidal Wave Functions (PSWFs).

These functions are also encountered in a different context. When solving the Helmholtz equation, $ΔΦ + k 2 Φ$ , by the method of separation of variables in prolate spheroidal coordinates, $(ξ,η,φ)$ , with:

$\ x=f \xi \eta,$

$\ y=f \sqrt{(\xi^2-1)(1-\eta^2)} \cos \phi,$

$\ z=f \sqrt{(\xi^2-1)(1-\eta^2)} \sin \phi,$

$\ |\xi|>1$ and

| η | < 1

the solution $Φ(ξ,η,φ)$ can be written as the product of a radial spheroidal wavefunction $R m n (c,ξ)$ and an angular spheroidal wavefunction $S m n (c,η)$ by $e i m φ$ with $c = f k / 2$ .

The radial wavefunction $R m n (c,ξ)$ satisfies the linear ordinary differential equation:

$\ (\xi^2 -1) \frac{d^2 R_{mn}(c,\xi)}{d^2 \xi} + 2\xi \frac{d R_{mn}(c,\xi)}{d \xi} -\left(\lambda_{mn}(c) -c^2 \xi^2 +\frac{m^2}{\xi^2-1}\right) {R_{mn}(c,\xi)} = 0$

The eigenvalue $λ m n (c)$ of this Sturm-Liouville differential equation is fixed by the requirement that $R m n (c,ξ)$ must be finite for $|\xi| \to 1_+$ .

The angular wavefunction satisfies the differential equation:

$\ (\eta^2 -1) \frac{d^2 S_{mn}(c,\eta)}{d^2 \eta} + 2\eta \frac{d S_{mn}(c,\eta)}{d \eta} -\left(\lambda_{mn}(c) -c^2 \eta^2 +\frac{m^2}{\eta^2-1}\right) {S_{mn}(c,\eta)} = 0$

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, $| ξ | > 1$ ) in the angular wavefunction $| η | < 1$ ).

For $c = 0$ these two differential equations reduce to the equations satisfied by the Legendre functions. For $c\ne 0$ , the angular spheroidal wavefunctions can be expanded as a series of Legendre functions.

Let us note that if one writes $S m n (c,η) = (1 - η 2) m / 2 Y m n (c,η)$ , the function $Y m n (c,η)$ satisfies the following linear ordinary differential equation:

$\ (1-\eta^2) \frac{d^2 Y_{mn}(c,\eta)}{d^2 \eta} -2 (m+1) \eta \frac{d Y_{mn}(c,\eta)}{d \eta} - +\left(c^2 \eta^2 +m(m+1)-\lambda_{mn}(c)\right) {Y_{mn}(c,\eta)} = 0,$

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.

[edit] References

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W. J. Thomson Spheroidal Wave functions Computing in Science & Engineering p. 84, May-June 1999.
I. Daubechies, "Ten Lectures on Wavelets"
C. Niven On the Conduction of Heat in Ellipsoids of Revolution. Philosophical transactions of the Royal Society of London, 171 p. 117 (1880)
J. A. Stratton Spheroidal functions Proceedings of the National Academy of Sciences (USA) 21, 51 (1935)
C. Flammer Spheroidal Wave Functions. Stanford, CA: Stanford University Press, 1957.
J. Meixner and F. W. Schäfke, Mathieusche Funktionen und Sphäroidfunktionen. Berlin: Springer-Verlag, 1954.
J. A. Stratton , P. M. Morse, J. L. Chu, J. D. C. Little, and F. J. Corbató, Spheroidal Wave Functions. New York: Wiley, 1956.
M. Abramowitz and I. Stegun Handbook of Mathematical Functions pp. 751-759 (Dover, New York, 1972)
F. Sleator Studies in Radar Cross-Sections -- XLIX. Diffraction and scattering by regular bodies III: the prolate spheroid (1964)
H. E. Hunter Tables of prolate spheroidal functions for m=0: Volume I. (1965)
H. E. Hunter Tables of prolate spheroidal functions for m=0 : Volume II. (1965)
P. E. Falloon (2002) Thesis on numerical computation of spheroidal functions University of Western Australia