- Prolate spheroidal wave functions
The prolate spheroidal wave functions are a set of functions derived by timelimiting and lowpassing, and a second timelimit operation. Let denote the time truncation operator, such that iff x is timelimited within . Similarly, let denote an ideal low-pass filtering operator, such that iff x is bandlimited within . The operator turns out to be linear,
bounded andself-adjoint . For we denote with the n-theigenfunction , defined as:
where are the associated eigenvalues. The timelimited functions are the Prolate Spheroidal Wave Functions (PSWFs).
These functions are also encountered in a different context. When solving the
Helmholtz equation ,, by the method of separation of variables inprolate spheroidal coordinates , , with::
:
:
: and . the solution can be writtenas the product of a radial spheroidal wavefunction and an angular spheroidal wavefunction by with .
The radial wavefunction satisfies the linear
ordinary differential equation ::
The eigenvalue of this Sturm-Liouville differential equation is fixed by the requirement that must be finite for .
The angular wavefunction satisfies the differential equation:
:
It is the same differential equation as in the case of the radial wavefunction. However, the range of the variable is different (in the radial wavefunction, ) in the angular wavefunction ).
For these two differential equations reduce to the equations satisfied by the
Legendre functions . For , the angular spheroidal wavefunctions can be expanded as a series of Legendre functions.Let us note that if one writes , the function satisfies the following linear ordinary differential equation:
which is known as the
spheroidal wave equation . This auxiliary equation is used for instance by Stratton 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 p. 758. Abramowitz and Stegun (and the present article) follow the notation of Flammer.
In the case of
oblate spheroidal coordinates the solution of the Helmholtz equation yieldsoblate spheroidal wavefunctions .Originally, the spheroidal wave functions were introduced by C. Niven in 1880 when studying the conduction of heat in an ellipsoid of revolution, which lead to a Helmholtz equation in spheroidal coordinates.
References
* W. J. Thomson [http://www.ece.nus.edu.sg/stfpage/elelilw/Software/00764220.pdf Spheroidal Wave functions] Computing in Science & Engineering p. 84, May-June 1999.
* I. Daubechies, "Ten Lectures on Wavelets"
* C. Niven " [http://gallica.bnf.fr/notice?N=FRBNF37571969 On the Conduction of Heat in Ellipsoids of Revolution.] " Philosophical transactions of the Royal Society of London, 171 p. 117 (1880)
* 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, 51 (1935)
* C. Flammer "Spheroidal Wave Functions." Stanford, CA: Stanford University Press, 1957.
* J. Meixner and F. W. Schäfke, "Mathieusche Funktionen und Sphäroidfunktionen." Berlin: Springer-Verlag, 1954.
* J. A. Stratton , P. M. Morse, J. L. Chu, J. D. C. Little, and F. J. Corbató, "Spheroidal Wave Functions." New York: Wiley, 1956.
* 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)
* F. Sleator " [http://hdl.handle.net/2027.42/21231 Studies in Radar Cross-Sections -- XLIX. Diffraction and scattering by regular bodies III: the prolate spheroid] " (1964)
* 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)
* P. E. Falloon (2002) [http://ftp.physics.uwa.edu.au/pub/Theses/2002/Falloon/Spheroidal/ Thesis on numerical computation of spheroidal functions] University of Western AustraliaExternal 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]
Wikimedia Foundation. 2010.