Prolate spheroidal wave function

Solutions to the wave equation

Solve the Helmholtz equation, \nabla^2 \Phi + k^2 \Phi=0, by the method of separation of variables in prolate spheroidal coordinates, (\xi,\eta,\varphi), with:

:\ x=a \sqrt{(\xi^2-1)(1-\eta^2)} \cos \varphi,

:\ y=a \sqrt{(\xi^2-1)(1-\eta^2)} \sin \varphi,

:\ z=a , \xi , \eta,

and \xi \ge 1, |\eta| \le 1 , and 0 \le \varphi \le 2\pi. Here, 2a 0 is the interfocal distance of the elliptical cross section of the prolate spheroid. Setting c=ka, the solution \Phi(\xi,\eta,\varphi) can be written as the product of e^{{\rm i} m \varphi}, a radial spheroidal wave function R_{mn}(c,\xi) and an angular spheroidal wave function S_{mn}(c,\eta).

The radial wave function R_{mn}(c,\xi) satisfies the linear ordinary differential equation:

:\ (\xi^2 -1) \frac{d^2 R_{mn}(c,\xi)}{d \xi ^2} + 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 angular wave function satisfies the differential equation:

:\ (1 - \eta^2) \frac{d^2 S_{mn}(c,\eta)}{d \eta ^2} - 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 wave function. However, the range of the variable is different: in the radial wave function, \xi \ge 1, while in the angular wave function, |\eta| \le 1. The eigenvalue \lambda_{mn}(c) of this Sturm–Liouville problem is fixed by the requirement that {S_{mn}(c,\eta)} must be finite for \eta \to \pm1.

For c=0 both differential equations reduce to the equations satisfied by the associated Legendre polynomials. For c\ne 0, the angular spheroidal wave functions can be expanded as a series of Legendre functions.

If one writes S_{mn}(c,\eta)=(1-\eta^2)^{m/2} Y_{mn}(c,\eta), the function Y_{mn}(c,\eta) satisfies

: \ (1-\eta^2) \frac{d^2 Y_{mn}(c,\eta)}{d \eta ^2} -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 has been used by Stratton.

Band-limited signals

In signal processing, the prolate spheroidal wave functions are useful as eigenfunctions of a time-limiting operation followed by a low-pass filter. Let D denote the time truncation operator, such that f(t)=D f(t) if and only if f(t) has support on [-T, T]. Similarly, let B denote an ideal low-pass filtering operator, such that f(t)=B f(t) if and only if its Fourier transform is limited to [-\Omega, \Omega]. The operator BD turns out to be linear, bounded and self-adjoint. For n=0,1,2,\ldots we denote by\psi_n(c,t) the n-th eigenfunction, defined as

: \ BD \psi_n(c,t) = \frac{1}{2\pi}\int_{-\Omega}^\Omega \left(\int_{-T}^T \psi_n(c,\tau)e^{-i\omega \tau} , d\tau\right)e^{i\omega t} , d\omega = \lambda_n(c)\psi_n(c,t),

where 1\lambda_0(c)\lambda_1(c)\cdots0 are the associated eigenvalues, and c=T\Omega is a constant. The band-limited functions {\psi_n(c,t)}{n=0}^{\infty} are the PSWFs, proportional to the S{0n}(c, t/T) introduced above. (See also Spectral concentration problem.)

Pioneering work in this area was performed by Slepian and Pollak, Landau and Pollak, and Slepian.

PSWFs whose domain is a (portion of) the surface of the unit sphere are more generally called "Slepian functions". These are of great utility in disciplines such as geodesy, cosmology, or tomography

Technical information and history

There are different normalization schemes for spheroidal functions. A table of the different schemes can be found in Abramowitz and Stegun who follow the notation of Flammer. The Digital Library of Mathematical Functions provided by NIST is an excellent resource for spheroidal wave functions.

Tables of numerical values of spheroidal wave functions are given in Flammer, Hanish et al., and Van Buren et al.

Originally, the spheroidal wave functions were introduced by C. Niven, which lead to a Helmholtz equation in spheroidal coordinates. Monographs tying together many aspects of the theory of spheroidal wave functions were written by Strutt, Stratton et al., Meixner and Schafke, and Flammer.

Flammer Patz and Van Buren, Baier et al., Zhang and Jin, Thompson and Falloon. Van Buren and Boisvert have recently developed new methods for calculating prolate spheroidal wave functions that extend the ability to obtain numerical values to extremely wide parameter ranges. Fortran source code that combines the new results with traditional methods is available at http://www.mathieuandspheroidalwavefunctions.com.

Asymptotic expansions of angular PSWFs for large values of c have been derived by Müller. He also investigated the relation between asymptotic expansions of spheroidal wave functions.

References

References

1. F.M. Arscott, ''Periodic Differential Equations'', Pergamon Press (1964).
2. J. A. Stratton ''[http://www.pnas.org/cgi/reprint/21/1/51?maxtoshow=&HITS=10&hits=10&RESULTFORMAT=1&title=spheroidal&andorexacttitle=and&andorexacttitleabs=and&andorexactfulltext=and&searchid=1&FIRSTINDEX=0&sortspec=relevance&resourcetype=HWCIT Spheroidal functions]'' Proceedings of the National Academy of Sciences (USA) '''21''' (1935) 51.
3. "''30.15 Spheroidal Wave Functions – Signal Analysis''". NIST.
4. D. Slepian and H. O. Pollak, ''[https://doi.org/10.1002/j.1538-7305.1961.tb03976.x Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty – I]'', Bell System Technical Journal '''40''' (1961) 43.
5. H. J. Landau and H. O. Pollak, ''[https://doi.org/10.1002/j.1538-7305.1961.tb03977.x Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty – II]'', Bell System Technical Journal '''40''' (1961) 65.
6. H. J. Landau and H. O. Pollak. ''[https://doi.org/10.1002/j.1538-7305.1962.tb03279.x Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty – III: The Dimension of the Space of Essentially Time- and Band-Limited Signals]'', Bell System Technical Journal '''41''' (1962) 1295.
7. D. Slepian ''[https://doi.org/10.1002/j.1538-7305.1964.tb01037.x Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty – IV: Extensions to Many Dimensions; Generalized Prolate Spheroidal Functions]'', Bell System Technical Journal '''43''' (1964) 3009–3057
8. D. Slepian. ''[https://doi.org/10.1002/j.1538-7305.1978.tb02104.x Prolate Spheroidal Wave Functions, Fourier Analysis, and Uncertainty – V: The Discrete Case]'', Bell System Technical Journal '''57''' (1978) 1371.
9. F. J. Simons, M. A. Wieczorek and F. A. Dahlen. ''Spatiospectral concentration on a sphere''. SIAM Review '''48''' (2006) 504–536, {{doi. 10.1137/S0036144504445765
10. F. J. Simons and F. A. Dahlen, ''Spherical Slepian functions and the polar gap in geodesy'', Geophysical Journal International '''166''' (2006) 1039–1061. {{doi. 10.1111/j.1365-246X.2006.03065.x
11. F. A. Dahlen and F. J. Simons, ''Spectral estimation on a sphere in geophysics and cosmology''. Geophysical Journal International '''174''' (2008) 774–807. {{doi. 10.1111/j.1365-246X.2008.03854.x
12. Marone F, Stampanoni M., ''Regridding reconstruction algorithm for real-time tomographic imaging''. J Synchrotron Radiat. (2012) {{ doi. 10.1107/S0909049512032864
13. M. Abramowitz and I. Stegun, ''Handbook of Mathematical Functions'' [http://www.math.sfu.ca/~cbm/aands/page_751.htm pp. 751–759] (Dover, New York, 1972)
14. C. Flammer, ''Spheroidal Wave Functions'', Stanford University Press, Stanford, CA, 1957.
15. Hunter,H. E. Hunter ''[http://hdl.handle.net/2027.42/5662 Tables of prolate spheroidal functions for m=0: Volume I.]'' (1965)
16. H. E. Hunter ''[http://hdl.handle.net/2027.42/5663 Tables of prolate spheroidal functions for m=0 : Volume II.]'' (1965)
17. (April 2025)
18. (April 2025)
19. (April 2025)
20. A. L. Van Buren, B. J. King, R. V. Baier, and S. Hanish. ''Tables of angular spheroidal wave functions, vol. 1, prolate, m = 0'', Naval Research Lab. Publication, U. S. Govt. Printing Office, 1975
21. C. Niven ''[http://gallica.bnf.fr/ark:/12148/bpt6k55976x.image.f135.tableDesMatieres.langEN On the conduction of heat in ellipsoids of revolution]'', Philosophical transactions of the Royal Society of London, '''171''' (1880) 117.
22. M. J. O. Strutt. ''Lamesche, Mathieusche and Verwandte Funktionen in Physik und Technik'', Ergebn. Math. u. Grenzgeb, '''1''' (1932) 199–323.
23. J. A. Stratton, P. M. Morse, J. L. Chu, and F. J. Corbató. ''Spheroidal Wave Functions'' Wiley, New York, 1956
24. J. Meixner and F. W. Schafke. ''Mathieusche Funktionen und Sphäroidfunktionen'', Springer-Verlag, Berlin, 1954
25. (April 2025)
26. R. V. Baier, A. L. Van Buren, S. Hanish, B. J. King – [https://dx.doi.org/10.1121/1.1974857 Spheroidal wave functions: their use and evaluation] The Journal of the Acoustical Society of America, '''48''' (1970) 102.
27. S. Zhang and J. Jin. ''Computation of Special Functions'', Wiley, New York, 1996
28. W. J. Thomson [http://www.ece.nus.edu.sg/stfpage/elelilw/Software/00764220.pdf Spheroidal Wave functions] {{webarchive. link. (2010-02-16 Computing in Science & Engineering p. 84, May–June 1999)
29. P. E. Falloon [http://ftp.physics.uwa.edu.au/pub/Theses/MSc/Falloon/Revised_Thesis.pdf Thesis on numerical computation of spheroidal functions] {{webarchive. link. (2011-04-11 University of Western Australia, 2002)
30. A. L. Van Buren and J. E. Boisvert. ''Accurate calculation of prolate spheroidal radial functions of the first kind and their first derivatives'', Quarterly of Applied Mathematics '''60''' (2002) 589-599.
31. A. L. Van Buren and J. E. Boisvert. ''Improved calculation of prolate spheroidal radial functions of the second kind and their first derivatives'', Quarterly of Applied Mathematics '''62''' (2004) 493–507.
32. H.J.W. Müller, ''Asymptotic Expansions of Prolate Spheroidal Wave Functions and their Characteristic Numbers'', J. reine u. angew. Math. '''212''' (1963) 26–48.
33. H.J.W. Müller, ''Asymptotische Entwicklungen von Sphäroidfunktionen und ihre Verwandtschaft mit Kugelfunktionen'', Z. angew. Math. Mech. '''44''' (1964) 371–374.
34. H.J.W. Müller, ''Über asymptotische Entwicklungen von Sphäroidfunktionen'', Z. angew. Math. Mech. '''45''' (1965) 29–36.