View source for Prolate spheroidal wave function

{{Short description|Special type of functions in mathematics}}
In [[mathematics]], '''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).<ref>F.M. Arscott, ''Periodic Differential Equations'', Pergamon Press (1964).</ref>

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

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

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

:<math>\ z=a \, \xi \, \eta, </math>

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

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

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

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

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

: <math>\ (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.<ref>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.</ref>

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

: <math>\ 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>1>\lambda_0(c)>\lambda_1(c)>\cdots>0</math> are the associated eigenvalues, and <math>c=T\Omega</math> is a constant. The band-limited functions <math>\{\psi_n(c,t)\}_{n=0}^{\infty}</math> are the prolate spheroidal wave functions, proportional to the <math>S_{0n}(c, t/T)</math> introduced above.<ref>{{cite web |title=''30.15 Spheroidal Wave Functions – Signal Analysis'' |url=https://dlmf.nist.gov/30.15 |website=Digital Library of Mathematical Functions |publisher=NIST |access-date=20 May 2021}}</ref> (See also [[Spectral concentration problem]].) 

Pioneering work in this area was performed by Slepian and Pollak,<ref>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.</ref> Landau and Pollak,<ref>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.</ref><ref>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.</ref> and Slepian.<ref>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</ref><ref>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.</ref>

Prolate spheroidal wave functions whose domain is a (portion of) the surface of the unit sphere are more generally called "Slepian functions".<ref>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}}</ref> These are of great utility in disciplines such as geodesy,<ref>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}}</ref> cosmology,<ref>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}}</ref> or tomography <ref>Marone F, Stampanoni M., ''Regridding reconstruction algorithm for real-time tomographic imaging''. J Synchrotron Radiat. (2012) {{ doi|10.1107/S0909049512032864}}</ref>

== 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<ref>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)</ref> who follow the notation of Flammer.<ref name=Flammer>C. Flammer, ''Spheroidal Wave Functions'', Stanford University Press, Stanford, CA, 1957.</ref>
The [https://dlmf.nist.gov/30 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,<ref name=Flammer/> Hunter,<ref>H. E. Hunter ''[http://hdl.handle.net/2027.42/5662 Tables of prolate spheroidal functions for m=0: Volume I.]'' (1965)</ref><ref>H. E. Hunter ''[http://hdl.handle.net/2027.42/5663 Tables of prolate spheroidal functions for m=0 : Volume II.]'' (1965)</ref> Hanish et al.,<ref>S. Hanish, R. V. Baier, A. L. Van Buren, and B. J. King ''[http://torpedo.nrl.navy.mil/tu/ps/doc.html?dsn=123401 Tables of radial spheroidal wave functions, volume 1, prolate, m = 0]'' (1970)</ref><ref>S. Hanish, R. V. Baier, A. L. Van Buren, and B. J. King ''[http://torpedo.nrl.navy.mil/tu/ps/doc.html?dsn=123163 Tables of radial spheroidal wave functions, volume 2, prolate, m = 1]'' (1970)</ref><ref>S. Hanish, R. V. Baier, A. L. Van Buren, and B. J. King ''[http://torpedo.nrl.navy.mil/tu/ps/doc.html?dsn=123165 Tables of radial spheroidal wave functions, volume 3, prolate, m = 2]'' (1970)</ref> and Van Buren et al.<ref>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</ref>

Originally, the spheroidal wave functions were introduced by C. Niven,<ref>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.</ref> 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,<ref>M. J. O. Strutt. ''Lamesche, Mathieusche and Verwandte Funktionen in Physik und Technik'', Ergebn. Math. u. Grenzgeb, '''1''' (1932) 199–323.</ref> Stratton et al.,<ref>J. A. Stratton, P. M. Morse, J. L. Chu, and F. J. Corbató. ''Spheroidal Wave Functions'' Wiley, New York, 1956</ref> Meixner and Schafke,<ref>J. Meixner and F. W. Schafke. ''Mathieusche Funktionen und Sphäroidfunktionen'', Springer-Verlag, Berlin, 1954</ref> and Flammer.<ref name=Flammer/>

Flammer<ref name=Flammer/> 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.,<ref>B. J. King, R. V. Baier, and S Hanish ''[http://torpedo.nrl.navy.mil/tu/ps/doc.html?dsn=124309 A Fortran computer program for calculating the prolate spheroidal radial functions of the first and second kind and their first derivatives.]'' (1970)</ref> Patz and Van Buren,<ref>B. J. Patz and A. L. Van Buren ''[http://torpedo.nrl.navy.mil/tu/ps/doc.html?dsn=354151 A Fortran computer program for calculating the prolate spheroidal angular functions of the first kind.]'' (1981)</ref> Baier et al.,<ref>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.</ref> Zhang and Jin,<ref>S. Zhang and J. Jin. ''Computation of Special Functions'', Wiley, New York, 1996</ref> Thompson<ref>W. J. Thomson [http://www.ece.nus.edu.sg/stfpage/elelilw/Software/00764220.pdf Spheroidal Wave functions] {{webarchive|url=https://web.archive.org/web/20100216213736/http://www.ece.nus.edu.sg/stfpage/elelilw/Software/00764220.pdf |date=2010-02-16 }} Computing in Science & Engineering p.&nbsp;84, May–June 1999</ref> and Falloon.<ref>P. E. Falloon [http://ftp.physics.uwa.edu.au/pub/Theses/MSc/Falloon/Revised_Thesis.pdf Thesis on numerical computation of spheroidal functions] {{webarchive|url=https://web.archive.org/web/20110411175924/http://ftp.physics.uwa.edu.au/pub/Theses/MSc/Falloon/Revised_Thesis.pdf |date=2011-04-11 }} University of Western Australia, 2002</ref> Van Buren and Boisvert<ref>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.</ref><ref>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.</ref> 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> c </math> have been derived by Müller.<ref>H.J.W. Müller, ''Asymptotic Expansions of Prolate Spheroidal Wave Functions and their Characteristic Numbers'', J. reine u. angew. Math. '''212''' (1963) 26–48.</ref> He also investigated the relation between asymptotic expansions of spheroidal wave functions.<ref>H.J.W. Müller, ''Asymptotische Entwicklungen von Sphäroidfunktionen und ihre Verwandtschaft mit Kugelfunktionen'', Z. angew. Math. Mech. '''44''' (1964) 371–374.</ref><ref>H.J.W. Müller, ''Über asymptotische Entwicklungen von Sphäroidfunktionen'', Z. angew. Math. Mech. '''45''' (1965) 29–36.</ref>

== References ==
{{Reflist}}

== External links ==
* MathWorld [http://mathworld.wolfram.com/SpheroidalWaveFunction.html Spheroidal Wave functions]
* MathWorld [http://mathworld.wolfram.com/ProlateSpheroidalWaveFunction.html Prolate Spheroidal Wave Function]
* MathWorld [http://mathworld.wolfram.com/OblateSpheroidalWaveFunction.html Oblate Spheroidal Wave function]

[[Category:Special functions]]
[[Category:Wavelets]]