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| In mathematics, the '''prolate spheroidal wave functions''' are a set of functions derived by timelimiting and lowpassing, and a second timelimit operation. Let <math>Q_T</math> denote the time truncation operator, such that <math>x=Q_T x</math> iff x is timelimited within <math>[-T/2;T/2]</math>. Similarly, let <math>P_W</math> denote an ideal low-pass filtering operator, such that <math>x=P_W x</math> iff x is bandlimited within <math>[-W;W]</math>. The operator <math>Q_T P_W Q_T</math> turns out to be linear, [[bounded function|bounded]] and [[self-adjoint]]. For <math>n=1,2,\ldots</math> we denote with <math>\psi_n</math> the n-th [[eigenfunction]], defined as
| | The author's title is Christy. Alaska is where I've always been living. What me and my family members love is to climb but I'm thinking on beginning some thing new. Office supervising is exactly where her main income comes from.<br><br>Also visit my website :: psychics online, [http://hknews.classicmall.com.hk/groups/some-simple-tips-for-personal-development-progress/ hknews.classicmall.com.hk], |
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| : <math>\ Q_T P_W Q_T \psi_n=\lambda_n\psi_n,</math>
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| where <math>\{\lambda_n\}_n</math> are the associated eigenvalues. The timelimited functions <math>\{\psi_n\}_n</math> are the Prolate Spheroidal Wave Functions (PSWFs). Pioneering work in this area was performed by Slepian and Pollak,<ref>D. Slepian and H. O. Pollak. ''[http://www3.alcatel-lucent.com/bstj/vol40-1961/articles/bstj40-1-43.pdf Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - I]'' Bell System Technical Journal '''40''' (1961)</ref> Landau and Pollak,<ref>H. J. Landau and H. O. Pollak. ''[http://www3.alcatel-lucent.com/bstj/vol40-1961/articles/bstj40-1-65.pdf Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - II]'' Bell System Technical Journal '''40''' (1961) | |
| </ref><ref>H. J. Landau and H. O. Pollak. ''[http://www3.alcatel-lucent.com/bstj/vol41-1962/articles/bstj41-4-1295.pdf 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)</ref> and Slepian.<ref>D. Slepian ''[http://www3.alcatel-lucent.com/bstj/vol43-1964/articles/bstj43-6-3009.pdf Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - IV: Extensions to Many Dimensions; Generalized Prolate Spheroidal Functions]'' Bell System Technical Journal '''43''' (1964)</ref><ref>D. Slepian. ''[http://www3.alcatel-lucent.com/bstj/vol57-1978/articles/bstj57-5-1371.pdf Prolate Spheroidal Wave Functions, Fourier Analysis, and Uncertainty - V: The Discrete Case]'' Bell System Technical Journal '''57''' (1978)</ref>
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| These functions are also encountered in a different context. When solving the [[Helmholtz equation]],
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| <math> \Delta \Phi + k^2 \Phi=0</math>, by the method of separation of variables in [[prolate spheroidal coordinates]], <math>(\xi,\eta,\phi)</math>, with:
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| :<math>\ x=(d/2) \xi \eta, </math>
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| :<math>\ y=(d/2) \sqrt{(\xi^2-1)(1-\eta^2)} \cos \phi, </math>
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| :<math>\ z=(d/2) \sqrt{(\xi^2-1)(1-\eta^2)} \sin \phi, </math>
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| :<math>\ \xi>=1 </math> and <math> |\eta|<=1 </math>.
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| the solution <math>\Phi(\xi,\eta,\phi)</math> can be written
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| as the product of a radial spheroidal wave function <math>R_{mn}(c,\xi)</math> and an angular spheroidal wave function <math>S_{mn}(c,\eta)</math> by <math>e^{i m \phi}</math>. Here <math>c=kd/2</math>, with <math>d</math> being the interfocal distance of the elliptical cross section of the prolate spheroid.
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| The radial wave function <math>R_{mn}(c,\xi)</math> satisfies the linear [[ordinary differential equation]]:
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| :<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>
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| The eigenvalue <math>\lambda_{mn}(c)</math> of this Sturm-Liouville differential equation is fixed by the requirement that <math>{R_{mn}(c,\xi)}</math> must be finite for <math> |\xi| \to 1_+</math>.
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| The angular wave function satisfies the differential equation:
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| :<math>\ (\eta^2 -1) \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>
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| 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>=1</math>) in the angular wave function <math>|\eta|<=1</math>).
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| For <math>c=0</math> these two 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.
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| Let us note that 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 the following linear ordinary differential equation:
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| <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>
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| which is known as the [[spheroidal wave equation]]. This auxiliary equation is used for instance 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''', 51 (1935)</ref> in his 1935 article.
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| 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> Abramowitz and Stegun (and the present article) follow the notation of Flammer.<ref name=Flammer>C. Flammer. ''Spheroidal Wave Functions'' Stanford University Press, Stanford, CA, 1957</ref>
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| 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''' p. 117 (1880)</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 Verdandte Funktionen in Physik und Technik'' Ergebn. Math. u. Grenzeb, '''1''', pp. 199-323, 1932</ref> Stratton et al.,<ref>J. A. Stratton, P. M. Morse, J. L. Chu, and F. J. Corbato. ''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/>
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| 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 - [http://dx.doi.org/10.1121/1.1974857 Spheroidal wave functions: their use and evaluation] The Journal of the Acoustical Society of America, '''48''', pp. 102–102 (1970)</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] Computing in Science & Engineering p. 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] 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 Mathemathics '''60''', pp. 589-599, 2002</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''', pp. 493-507, 2004</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.
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| 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>
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| The Digital Library of Mathematical Functions http://dlmf.nist.gov provided by NIST is an excellent resource for spheroidal wave functions.
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| 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, 2006, {{doi|10.1137/S0036144504445765}}</ref> (see also [[Spectral concentration problem]]). These are of great utility in disciplines such as geodesy<ref>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}}</ref> or cosmology.<ref>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}}</ref>
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| == References ==
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| <references/>
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| == External links ==
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| * MathWorld [http://mathworld.wolfram.com/SpheroidalWaveFunction.html Spheroidal Wave functions]
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| * MathWorld [http://mathworld.wolfram.com/ProlateSpheroidalWaveFunction.html Prolate Spheroidal Wave Function]
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| * MathWorld [http://mathworld.wolfram.com/OblateSpheroidalWaveFunction.html Oblate Spheroidal Wave function]
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| [[Category:Special functions]]
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| [[Category:Wavelets]]
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The author's title is Christy. Alaska is where I've always been living. What me and my family members love is to climb but I'm thinking on beginning some thing new. Office supervising is exactly where her main income comes from.
Also visit my website :: psychics online, hknews.classicmall.com.hk,