Prolate spheroidal wave function

In mathematics, the Prolate spheroidal wave functions (PSWF) are a set of orthogonal bandlimited functions. They are eigenfunctions of a timelimiting operation followed by a lowpassing operation. Let ${\displaystyle D}$ denote the time truncation operator, such that ${\displaystyle f(t)=Df(t)}$ if and only if ${\displaystyle f(t)}$ is timelimited within ${\displaystyle [-T/2;T/2]}$. Similarly, let ${\displaystyle B}$ denote an ideal low-pass filtering operator, such that ${\displaystyle f(t)=Bf(t)}$ if and only if ${\displaystyle f(t)}$ is bandlimited within ${\displaystyle [-\Omega ;\Omega ]}$. The operator ${\displaystyle BD}$ turns out to be linear, bounded and self-adjoint. For ${\displaystyle n=0,1,2,\ldots }$ we denote with ${\displaystyle \psi _{n}(c,t)}$ the n-th eigenfunction, defined as

${\displaystyle \ BD\psi _{n}(c,t)={\frac {1}{2\pi }}\int _{-\Omega }^{\Omega }\left(\int _{-T/2}^{T/2}\psi _{n}(c,\tau )e^{-i\omega \tau }d\tau \right)e^{i\omega t}d\omega =\lambda _{n}(c)\psi _{n}(c,t),}$

where ${\displaystyle 1>\lambda _{0}(c)>\lambda _{1}(c)>\ldots >0}$ are the associated eigenvalues, and ${\displaystyle 2c=T\Omega }$ is a constant. The timelimited functions ${\displaystyle \{\psi _{n}(c,t)\}_{n=0}^{\infty }}$ are the Prolate Spheroidal Wave Functions (PSWFs). Pioneering work in this area was performed by Slepian and Pollak,[1] Landau and Pollak,[2][3] and Slepian.[4][5]

These functions are also encountered in a different context. When solving the Helmholtz equation, ${\displaystyle \Delta \Phi +k^{2}\Phi =0}$, by the method of separation of variables in prolate spheroidal coordinates, ${\displaystyle (\xi ,\eta ,\phi )}$, with:

${\displaystyle \ x=(d/2)\xi \eta ,}$
${\displaystyle \ y=(d/2){\sqrt {(\xi ^{2}-1)(1-\eta ^{2})}}\cos \phi ,}$
${\displaystyle \ z=(d/2){\sqrt {(\xi ^{2}-1)(1-\eta ^{2})}}\sin \phi ,}$
${\displaystyle \ \xi >=1}$ and ${\displaystyle |\eta |<=1}$.

the solution ${\displaystyle \Phi (\xi ,\eta ,\phi )}$ can be written as the product of a radial spheroidal wave function ${\displaystyle R_{mn}(c,\xi )}$ and an angular spheroidal wave function ${\displaystyle S_{mn}(c,\eta )}$ by ${\displaystyle e^{im\phi }}$. Here ${\displaystyle c=kd/2}$, with ${\displaystyle d}$ being the interfocal distance of the elliptical cross section of the prolate spheroid.

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

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

The eigenvalue ${\displaystyle \lambda _{mn}(c)}$ of this Sturm-Liouville differential equation is fixed by the requirement that ${\displaystyle {R_{mn}(c,\xi )}}$ must be finite for ${\displaystyle |\xi |\to 1_{+}}$.

The angular wave function satisfies the differential equation:

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

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, ${\displaystyle \xi >=1}$) in the angular wave function ${\displaystyle |\eta |<=1}$).

For ${\displaystyle c=0}$ these two differential equations reduce to the equations satisfied by the associated Legendre polynomials. For ${\displaystyle c\neq 0}$, the angular spheroidal wave functions can be expanded as a series of Legendre functions.

Let us note that if one writes ${\displaystyle S_{mn}(c,\eta )=(1-\eta ^{2})^{m/2}Y_{mn}(c,\eta )}$, the function ${\displaystyle Y_{mn}(c,\eta )}$ satisfies the following linear ordinary differential equation:

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

which is known as the spheroidal wave equation. This auxiliary equation is used for instance by Stratton[6] 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.[7] Abramowitz and Stegun (and the present article) follow the notation of Flammer.[8]

Originally, the spheroidal wave functions were introduced by C. Niven,[9] 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,[10] Stratton et al.,[11] Meixner and Schafke,[12] and Flammer.[8]

Flammer[8] 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.,[13] Patz and Van Buren,[14] Baier et al.,[15] Zhang and Jin,[16] Thompson,[17] and Falloon.[18] Van Buren and Boisvert[19][20] 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.

Tables of numerical values of spheroidal wave functions are given in Flammer,[8] Hunter,[21][22] Hanish et al.,[23][24][25] and Van Buren et al.[26]

The Digital Library of Mathematical Functions http://dlmf.nist.gov provided by NIST is an excellent resource for spheroidal wave functions.

Prolate spheroidal wave functions whose domain is a (portion of) the surface of the unit sphere are more generally called "Slepian functions"[27] (see also Spectral concentration problem). These are of great utility in disciplines such as geodesy[28] or cosmology.[29]

Asymptotic expansions of angular prolate spheroidal wave functions for large values of ${\displaystyle c}$ have been derived by Müller.[30] The relation between asymptotic expansions of spheroidal wave functions and Legendre functions has been investigated in Müller [31] and Müller.[32]

References

1. D. Slepian and H. O. Pollak. Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - I Bell System Technical Journal 40 (1961)
2. H. J. Landau and H. O. Pollak. Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - II Bell System Technical Journal 40 (1961)
3. 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)
4. 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)
5. D. Slepian. Prolate Spheroidal Wave Functions, Fourier Analysis, and Uncertainty - V: The Discrete Case Bell System Technical Journal 57 (1978)
6. J. A. Stratton Spheroidal functions Proceedings of the National Academy of Sciences (USA) 21, 51 (1935)
7. . M. Abramowitz and I. Stegun. Handbook of Mathematical Functions pp. 751-759 (Dover, New York, 1972)
8. C. Flammer. Spheroidal Wave Functions Stanford University Press, Stanford, CA, 1957
9. C. Niven )On the conduction of heat in ellipsoids of revolution. Philosophical transactions of the Royal Society of London, 171 p. 117 (1880)
10. M. J. O. Strutt. Lamesche, Mathieusche and Verdandte Funktionen in Physik und Technik Ergebn. Math. u. Grenzeb, 1, pp. 199-323, 1932
11. J. A. Stratton, P. M. Morse, J. L. Chu, and F. J. Corbató. Spheroidal Wave Functions Wiley, New York, 1956
12. J. Meixner and F. W. Schafke. Mathieusche Funktionen und Sphäroidfunktionen Springer-Verlag, Berlin, 1954
13. 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)
14. B. J. Patz and A. L. Van Buren A Fortran computer program for calculating the prolate spheroidal angular functions of the first kind. (1981)
15. 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, pp. 102–102 (1970)
16. S. Zhang and J. Jin. Computation of Special Functions, Wiley, New York, 1996
17. W. J. Thomson Spheroidal Wave functions Archived 2010-02-16 at the Wayback Machine Computing in Science & Engineering p. 84, May–June 1999
18. P. E. Falloon Thesis on numerical computation of spheroidal functions Archived 2011-04-11 at the Wayback Machine University of Western Australia, 2002
19. 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 Mathemathics 60, pp. 589-599, 2002
20. 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, pp. 493-507, 2004
21. H. E. Hunter Tables of prolate spheroidal functions for m=0: Volume I. (1965)
22. H. E. Hunter Tables of prolate spheroidal functions for m=0 : Volume II. (1965)
23. 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)
24. 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)
25. 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)
26. 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
27. F. J. Simons, M. A. Wieczorek and F. A. Dahlen. Spatiospectral concentration on a sphere. SIAM Review, 2006, doi:10.1137/S0036144504445765
28. F. J. Simons and Dahlen, F. A. Spherical Slepian functions and the polar gap in Geodesy. Geophysical Journal International, 2006, doi:10.1111/j.1365-246X.2006.03065.x
29. F. A. Dahlen and F. J. Simons. Spectral estimation on a sphere in geophysics and cosmology. Geophysical Journal International, 2008, doi:10.1111/j.1365-246X.2008.03854.x
30. H.J.W. Müller, Asymptotic Expansions of Prolate Spheroidal Wave Functions and their Characteristic Numbers, J. reine u. angew. Math. 212 (1963) 26 - 48
31. H.J.W. Müller, Asymptotische Entwicklungen von Sphäroidfunktionen und ihre Verwandtschaft mit Kugelfunktionen, Z. angew. Math. Mech. 44 (1964) 371 - 374.
32. H.J.W. Müller, Über asymptotische Entwicklungen von Sphäroidfunktionen, Z. angew. Math. Mech. 45 (1965) 29 - 36.