Prolate spheroidal wave function

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The prolate spheroidal wave functions are eigenfunctions 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 (“pancake shaped” ellipsoid).[1]

Solutions to the wave equation

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

[math]\displaystyle{ \ x=a \sqrt{(\xi^2-1)(1-\eta^2)} \cos \varphi, }[/math]
[math]\displaystyle{ \ y=a \sqrt{(\xi^2-1)(1-\eta^2)} \sin \varphi, }[/math]
[math]\displaystyle{ \ z=a \, \xi \, \eta, }[/math]

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

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

[math]\displaystyle{ \ (\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 }[/math]

The angular wave function satisfies the differential equation:

[math]\displaystyle{ \ (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 }[/math]

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, [math]\displaystyle{ \xi \ge 1 }[/math], while in the angular wave function, [math]\displaystyle{ |\eta| \le 1 }[/math]. The eigenvalue [math]\displaystyle{ \lambda_{mn}(c) }[/math] of this Sturm–Liouville problem is fixed by the requirement that [math]\displaystyle{ {S_{mn}(c,\eta)} }[/math] must be finite for [math]\displaystyle{ \eta \to \pm1 }[/math].

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

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

[math]\displaystyle{ \ (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, }[/math]

which is known as the spheroidal wave equation. This auxiliary equation has been used by Stratton.[2]

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 [math]\displaystyle{ D }[/math] denote the time truncation operator, such that [math]\displaystyle{ f(t)=D f(t) }[/math] if and only if [math]\displaystyle{ f(t) }[/math] has support on [math]\displaystyle{ [-T, T] }[/math]. Similarly, let [math]\displaystyle{ B }[/math] denote an ideal low-pass filtering operator, such that [math]\displaystyle{ f(t)=B f(t) }[/math] if and only if its Fourier transform is limited to [math]\displaystyle{ [-\Omega, \Omega] }[/math]. The operator [math]\displaystyle{ BD }[/math] turns out to be linear, bounded and self-adjoint. For [math]\displaystyle{ n=0,1,2,\ldots }[/math] we denote with [math]\displaystyle{ \psi_n(c,t) }[/math] the [math]\displaystyle{ n }[/math]-th eigenfunction, defined as

[math]\displaystyle{ \ 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), }[/math]

where [math]\displaystyle{ 1\gt \lambda_0(c)\gt \lambda_1(c)\gt \cdots\gt 0 }[/math] are the associated eigenvalues, and [math]\displaystyle{ c=T\Omega }[/math] is a constant. The band-limited functions [math]\displaystyle{ \{\psi_n(c,t)\}_{n=0}^{\infty} }[/math] are the prolate spheroidal wave functions, proportional to the [math]\displaystyle{ S_{0n}(c, t/T) }[/math] introduced above.[3] (See also Spectral concentration problem.)

Pioneering work in this area was performed by Slepian and Pollak,[4] Landau and Pollak,[5][6] and Slepian.[7][8]

Prolate spheroidal wave functions whose domain is a (portion of) the surface of the unit sphere are more generally called "Slepian functions".[9] These are of great utility in disciplines such as geodesy[10] or cosmology.[11]

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[12] who follow the notation of Flammer.[13] 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,[13] Hunter,[14][15] Hanish et al.,[16][17][18] and Van Buren et al.[19]

Originally, the spheroidal wave functions were introduced by C. Niven,[20] 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,[21] Stratton et al.,[22] Meixner and Schafke,[23] and Flammer.[13]

Flammer[13] provided a thorough discussion of the calculation of the eigenvalues, angular wavefunctions, and radial wavefunctions for both the prolate and the oblate case. Computer programs for this purpose have been developed by many, including King et al.,[24] Patz and Van Buren,[25] Baier et al.,[26] Zhang and Jin,[27] Thompson[28] and Falloon.[29] Van Buren and Boisvert[30][31] 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 prolate spheroidal wave functions for large values of [math]\displaystyle{ c }[/math] have been derived by Müller.[32] He also investigated the relation between asymptotic expansions of spheroidal wave functions.[33][34]

References

  1. ↑ F.M. Arscott, Periodic Differential Equations, Pergamon Press (1964).
  2. ↑ J. A. Stratton Spheroidal functions Proceedings of the National Academy of Sciences (USA) 21 (1935) 51.
  3. ↑ "30.15 Spheroidal Wave Functions – Signal Analysis". NIST. https://dlmf.nist.gov/30.15. 
  4. ↑ D. Slepian and H. O. Pollak, Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty – I, Bell System Technical Journal 40 (1961) 43.
  5. ↑ H. J. Landau and H. O. Pollak, Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty – II, Bell System Technical Journal 40 (1961) 65.
  6. ↑ H. J. Landau and H. O. Pollak. 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 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. 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. ↑ M. Abramowitz and I. Stegun, Handbook of Mathematical Functions pp. 751–759 (Dover, New York, 1972)
  13. ↑ 13.0 13.1 13.2 13.3 C. Flammer, Spheroidal Wave Functions, Stanford University Press, Stanford, CA, 1957.
  14. ↑ H. E. Hunter Tables of prolate spheroidal functions for m=0: Volume I. (1965)
  15. ↑ H. E. Hunter Tables of prolate spheroidal functions for m=0 : Volume II. (1965)
  16. ↑ S. Hanish, R. V. Baier, A. L. Van Buren, and B. J. King Tables of radial spheroidal wave functions, volume 1, prolate, m = 0 (1970)
  17. ↑ S. Hanish, R. V. Baier, A. L. Van Buren, and B. J. King Tables of radial spheroidal wave functions, volume 2, prolate, m = 1 (1970)
  18. ↑ S. Hanish, R. V. Baier, A. L. Van Buren, and B. J. King Tables of radial spheroidal wave functions, volume 3, prolate, m = 2 (1970)
  19. ↑ 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
  20. ↑ C. Niven On the conduction of heat in ellipsoids of revolution, Philosophical transactions of the Royal Society of London, 171 (1880) 117.
  21. ↑ M. J. O. Strutt. Lamesche, Mathieusche and Verwandte Funktionen in Physik und Technik, Ergebn. Math. u. Grenzgeb, 1 (1932) 199–323.
  22. ↑ J. A. Stratton, P. M. Morse, J. L. Chu, and F. J. Corbató. Spheroidal Wave Functions Wiley, New York, 1956
  23. ↑ J. Meixner and F. W. Schafke. Mathieusche Funktionen und Sphäroidfunktionen, Springer-Verlag, Berlin, 1954
  24. ↑ B. J. King, R. V. Baier, and S Hanish A Fortran computer program for calculating the prolate spheroidal radial functions of the first and second kind and their first derivatives. (1970)
  25. ↑ B. J. Patz and A. L. Van Buren A Fortran computer program for calculating the prolate spheroidal angular functions of the first kind. (1981)
  26. ↑ R. V. Baier, A. L. Van Buren, S. Hanish, B. J. King – 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 Spheroidal Wave functions Computing in Science & Engineering p. 84, May–June 1999
  29. ↑ P. E. Falloon Thesis on numerical computation of spheroidal functions 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.

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