查看“︁長球波函數”︁的源代码

在數學中，'''長球波函數'''由一個限時、限頻、與第二個限時的函數相乘而成。假定<math>Q_T</math>表示一個切截時間的運算器，且<math>x=Q_T x</math>，則x必為有限時間區間的函數，當x在<math>[-T/2;T/2]</math>的區間內。同理，假定<math>P_W</math>表示一個理想的低頻濾波器，且<math>x=P_W x</math>，則x必為有限頻寬區間的函數，當x在<math>[-W;W]</math>的區間內。透過組合上述運算子，使得<math>Q_T P_W Q_T</math>轉變成線性、[[有界算子|有界]]且{{le|自伴|Self-adjoint}}的運算式。對於<math>n=1,2,\ldots</math>，我們假設<math>\psi_n</math>為第n項的[[本徵函數]]，定義下列函式

: <math>\ Q_T P_W Q_T \psi_n=\lambda_n\psi_n,</math>

其中<math>\{\lambda_n\}_n</math>為對應的本徵值。此時限函數<math>\{\psi_n\}_n</math>即為長球波函數(PSWFs).在此領域中，幾個非常重要的先驅文章由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]{{dead link|date=2018年5月 |bot=InternetArchiveBot |fix-attempted=yes }}'' 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]{{dead link|date=2018年5月 |bot=InternetArchiveBot |fix-attempted=yes }}''  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]{{dead link|date=2018年5月 |bot=InternetArchiveBot |fix-attempted=yes }}'' 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]{{dead link|date=2018年5月 |bot=InternetArchiveBot |fix-attempted=yes }}'' 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]{{dead link|date=2018年5月 |bot=InternetArchiveBot |fix-attempted=yes }}''  Bell System Technical Journal '''57''' (1978)</ref>所提出。

這些函數有些不同的意涵，當在解[[亥姆霍兹方程]]<math> \Delta \Phi + k^2 \Phi=0</math>，透過在[[長球面坐標系]]做變數分離，使得<math>(\xi,\eta,\phi)</math>各代表：

:<math>\ x=(d/2) \xi \eta, </math>

:<math>\ y=(d/2) \sqrt{(\xi^2-1)(1-\eta^2)} \cos \phi, </math>

:<math>\ z=(d/2) \sqrt{(\xi^2-1)(1-\eta^2)} \sin \phi, </math>

:<math>\ \xi>=1 </math>  and <math> |\eta|<=1 </math>.
 
得到解<math>\Phi(\xi,\eta,\phi)</math>為長球波函數<math>R_{mn}(c,\xi)</math>與角球波函數<math>S_{mn}(c,\eta)</math>的成積乘上<math>e^{i m \phi}</math>. 這裡的變數c可定義為<math>c=kd/2</math>, with <math>d</math>為長球的橢圓截面的兩焦點的距離。

徑向波(The radial wave function)<math>R_{mn}(c,\xi)</math>滿足線性[[常微分方程]]:

:<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>

此本徵值<math>\lambda_{mn}(c)</math>在[[施图姆-刘维尔理论|Sturm-Liouville 微分方程]]中已被固定，透過設定<math>{R_{mn}(c,\xi)}</math>為有限函數，當 <math> |\xi| \to 1_+</math>.

角波函數滿足下列微分方程：

:<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>

這跟徑向波函數式為同樣的微分方程。然而，這兩式的變數的範圍是不同的(在徑向波函數中<math>\xi>=1</math>)，在角波函數中<math>|\eta|<=1</math>)。

當給定<math>c=0</math>，這兩個微分方程可以簡化成滿足[[伴隨勒讓德多項式]]的式子。當給定<math>c\ne 0</math>，角波函數可被展開成勒讓德級數。

注意，如果我們將角波函數寫成<math>S_{mn}(c,\eta)=(1-\eta^2)^{m/2} Y_{mn}(c,\eta)</math>，函數<math>Y_{mn}(c,\eta)</math>將滿足以下線性微分方程：

<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>

此函數為球波函數。這個輔助方程式在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] {{Wayback|url=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 |date=20190710184216 }}'' Proceedings of the National Academy of Sciences (USA) '''21''', 51 (1935)</ref> 1935年的文章被當作例子。

現存不少不同的球函數標準化的方法，在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] {{Wayback|url=http://www.math.sfu.ca/~cbm/aands/page_751.htm |date=20100728061534 }} (Dover, New York, 1972)</ref>的文章中有整理的表格。Abramowitz跟Stegun(以及現在的相關文章)都沿用Flammer當初提出來的符號<ref name=Flammer>C. Flammer. ''Spheroidal Wave Functions'' Stanford University Press, Stanford, CA, 1957</ref>。

一開始，球波函數是由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.] {{Wayback|url=http://gallica.bnf.fr/ark:/12148/bpt6k55976x.image.f135.tableDesMatieres.langEN |date=20190710184218 }}'' Philosophical transactions of the Royal Society of London, '''171''' p.&nbsp;117 (1880)</ref>提出，他在球座標上引入Helmholtz方程式。許多專題論文已經探討出球波函數的很多面向，例如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/>等人的作品。

Flammer<ref name=Flammer/>提供了一個完整的討論，計算出長球與扁球的本徵值、角波函數與徑函數。許多電腦程式已經因應發展出來，其中包含King與其團隊,<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.] {{Wayback|url=http://torpedo.nrl.navy.mil/tu/ps/doc.html?dsn=124309 |date=20190710184248 }}'' (1970)</ref> Patz和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.] {{Wayback|url=http://torpedo.nrl.navy.mil/tu/ps/doc.html?dsn=354151 |date=20190710184218 }}'' (1981)</ref> Baier與其團隊,<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] {{Wayback|url=http://dx.doi.org/10.1121/1.1974857 |date=20190710184217 }} The Journal of the Acoustical Society of America, '''48''', pp.&nbsp;102–102 (1970)</ref> Zhang和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>、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和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和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>最近發展出新的方法去計算出長球波函數，延伸了數值解的能力，能運算極廣的變數範圍。Fortran原始碼結合了新的結果與傳統的方法，可見於http://www.mathieuandspheroidalwavefunctions.com.。 {{Wayback|url=http://www.mathieuandspheroidalwavefunctions.com./ |date=20181229222641 }}
 
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.] {{Wayback|url=http://hdl.handle.net/2027.42/5662 |date=20190710184219 }}'' (1965)</ref><ref>H. E. Hunter ''[http://hdl.handle.net/2027.42/5663 Tables of prolate spheroidal functions for m=0 : Volume II.] {{Wayback|url=http://hdl.handle.net/2027.42/5663 |date=20190710184218 }}'' (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]{{Dead link}}'' (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] {{Wayback|url=http://torpedo.nrl.navy.mil/tu/ps/doc.html?dsn=123163 |date=20190710184218 }}'' (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] {{Wayback|url=http://torpedo.nrl.navy.mil/tu/ps/doc.html?dsn=123165 |date=20190710184219 }}'' (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>等人也提出了數值解的整理表格。

NIST提供的DLMF(Digital Library of Mathematical Functions)(http://dlmf.nist.gov)是個了解球波函數的良好資源。{{Dead link|date=2018年6月 |bot=InternetArchiveBot |fix-attempted=no }}

關於值域落在單位球的表面的長球波函數，我們通稱為"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> (另見「頻譜集中問題」)。這函數存在非常多的應用，像是大地測量<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>以及宇宙學.<ref>F. A. Dahlen和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>

== 參考文獻 ==
{{reflist}}

== 外部連結 ==
* MathWorld [http://mathworld.wolfram.com/SpheroidalWaveFunction.html Spheroidal Wave functions] {{Wayback|url=http://mathworld.wolfram.com/SpheroidalWaveFunction.html |date=20200106094646 }}
* MathWorld [http://mathworld.wolfram.com/ProlateSpheroidalWaveFunction.html Prolate Spheroidal Wave Function] {{Wayback|url=http://mathworld.wolfram.com/ProlateSpheroidalWaveFunction.html |date=20210520084805 }}
* MathWorld [http://mathworld.wolfram.com/OblateSpheroidalWaveFunction.html Oblate Spheroidal Wave function] {{Wayback|url=http://mathworld.wolfram.com/OblateSpheroidalWaveFunction.html |date=20210520084723 }}

[[Category:特殊函数]]
[[Category:小波分析]]