prolate spheroidal wave function

Solve the Helmholtz equation,

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

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

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

:$\backslash \; z=a\; \backslash ,\; \backslash xi\; \backslash ,\; \backslash eta,$

and $\backslash xi\; \backslash ge\; 1$, $|\backslash eta|\; \backslash le\; 1$, and $0\; \backslash le\; \backslash varphi\; \backslash le\; 2\backslash pi$. Here, $2a\; >\; 0$ is the interfocal distance of the elliptical cross section of the prolate spheroid.

Setting $c=ka$, the solution $\backslash Phi(\backslash xi,\backslash eta,\backslash varphi)$ can be written

as the product of $e^\{\{\backslash rm\; i\}\; m\; \backslash varphi\}$, a radial spheroidal wave function $R\_\{mn\}(c,\backslash xi)$ and an angular spheroidal wave function $S\_\{mn\}(c,\backslash eta)$.

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

:$\backslash \; (\backslash xi^2\; -1)\; \backslash frac\{d^2\; R\_\{mn\}(c,\backslash xi)\}\{d\; \backslash xi\; ^2\}\; +\; 2\backslash xi\; \backslash frac\{d\; R\_\{mn\}(c,\backslash xi)\}\{d\; \backslash xi\}\; -\backslash left(\backslash lambda\_\{mn\}(c)\; -c^2\; \backslash xi^2\; +\backslash frac\{m^2\}\{\backslash xi^2-1\}\backslash right)\; \{R\_\{mn\}(c,\backslash xi)\}\; =\; 0$

The angular wave function satisfies the differential equation:

:$\backslash \; (1\; -\; \backslash eta^2)\; \backslash frac\{d^2\; S\_\{mn\}(c,\backslash eta)\}\{d\; \backslash eta\; ^2\}\; -\; 2\backslash eta\; \backslash frac\{d\; S\_\{mn\}(c,\backslash eta)\}\{d\; \backslash eta\}\; +\backslash left(\backslash lambda\_\{mn\}(c)\; -c^2\; \backslash eta^2\; +\backslash frac\{m^2\}\{\backslash eta^2-1\}\backslash right)\; \{S\_\{mn\}(c,\backslash eta)\}\; =\; 0$

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, $\backslash xi\; \backslash ge\; 1$, while in the angular wave function, $|\backslash eta|\; \backslash le\; 1$. The eigenvalue $\backslash lambda\_\{mn\}(c)$ of this Sturm–Liouville problem is fixed by the requirement that $\{S\_\{mn\}(c,\backslash eta)\}$ must be finite for $\backslash eta\; \backslash to\; \backslash pm1$.

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

If one writes $S\_\{mn\}(c,\backslash eta)=(1-\backslash eta^2)^\{m/2\}\; Y\_\{mn\}(c,\backslash eta)$, the function $Y\_\{mn\}(c,\backslash eta)$ satisfies

: $\backslash \; (1-\backslash eta^2)\; \backslash frac\{d^2\; Y\_\{mn\}(c,\backslash eta)\}\{d\; \backslash eta\; ^2\}\; -2\; (m+1)\; \backslash eta\; \backslash frac\{d\; Y\_\{mn\}(c,\backslash eta)\}\{d\; \backslash eta\}\; -\; \backslash left(c^2\; \backslash eta^2\; +m(m+1)-\backslash lambda\_\{mn\}(c)\backslash right)\; \{Y\_\{mn\}(c,\backslash eta)\}\; =\; 0,$

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

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 $D$ 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 $[-\backslash Omega,\; \backslash Omega]$. The operator $BD$ turns out to be linear, bounded and self-adjoint. For $n=0,1,2,\backslash ldots$ we denote with $\backslash psi\_n(c,t)$ the $n$-th eigenfunction, defined as

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

where $1>\backslash lambda\_0(c)>\backslash lambda\_1(c)>\backslash cdots>0$ are the associated eigenvalues, and $c=T\backslash Omega$ is a constant. The band-limited functions $\backslash \{\backslash psi\_n(c,t)\backslash \}\_\{n=0\}^\{\backslash infty\}$ are the prolate spheroidal wave functions, proportional to the $S\_\{0n\}(c,\; t/T)$ introduced above.{{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}} (See also Spectral concentration problem.)

Pioneering work in this area was performed by Slepian and Pollak,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. Landau and Pollak,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.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. and Slepian.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) 3009–3057D. 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.

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

There are different normalization schemes for spheroidal functions. A table of the different schemes can be found in Abramowitz and StegunM. 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) who follow the notation of Flammer.C. Flammer, Spheroidal Wave Functions, Stanford University Press, Stanford, CA, 1957.

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, Hunter,H. E. Hunter [http://hdl.handle.net/2027.42/5662 Tables of prolate spheroidal functions for m=0: Volume I.] (1965)H. E. Hunter [http://hdl.handle.net/2027.42/5663 Tables of prolate spheroidal functions for m=0 : Volume II.] (1965) Hanish et al.,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]{{dead link|date=April 2025|bot=medic}}{{cbignore|bot=medic}} (1970)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]{{dead link|date=April 2025|bot=medic}}{{cbignore|bot=medic}} (1970)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]{{dead link|date=April 2025|bot=medic}}{{cbignore|bot=medic}} (1970) and Van Buren et al.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

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

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.,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.]{{dead link|date=April 2025|bot=medic}}{{cbignore|bot=medic}} (1970) Patz and Van Buren,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.]{{dead link|date=April 2025|bot=medic}}{{cbignore|bot=medic}} (1981) Baier et al.,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. Zhang and Jin,S. Zhang and J. Jin. Computation of Special Functions, Wiley, New York, 1996 ThompsonW. 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. 84, May–June 1999 and Falloon.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 Van Buren and BoisvertA. 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.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. 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 $c$ have been derived by Müller.H.J.W. Müller, Asymptotic Expansions of Prolate Spheroidal Wave Functions and their Characteristic Numbers, J. reine u. angew. Math. 212 (1963) 26–48. He also investigated the relation between asymptotic expansions of spheroidal wave functions.H.J.W. Müller, Asymptotische Entwicklungen von Sphäroidfunktionen und ihre Verwandtschaft mit Kugelfunktionen, Z. angew. Math. Mech. 44 (1964) 371–374.H.J.W. Müller, Über asymptotische Entwicklungen von Sphäroidfunktionen, Z. angew. Math. Mech. 45 (1965) 29–36.

{{Reflist}}

• 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]

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