長球波函數

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。假定表示一個切截時間的運算器，且，則x必為有限時間區間的函數，當x在的區間內。同理，假定表示一個理想的低頻濾波器，且，則x必為有限頻寬區間的函數，當x在的區間內。透過組合上述運算子，使得轉變成線性、有界且的運算式。對於，我們假設為第n項的本徵函數，定義下列函式



其中為對應的本徵值。此時限函數即為長球波函數(PSWFs).在此領域中，幾個非常重要的先驅文章由Slepian and Pollak,D. Slepian and H. O. Pollak. Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - I Bell System Technical Journal 40 (1961) Landau and Pollak,H. J. Landau and H. O. Pollak. Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - II  Bell System Technical Journal 40 (1961)H. J. Landau and H. O. Pollak. 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) and Slepian.D. Slepian Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - IV: Extensions to Many Dimensions; Generalized Prolate Spheroidal Functions Bell System Technical Journal 43 (1964)D. Slepian. Prolate Spheroidal Wave Functions, Fourier Analysis, and Uncertainty - V: The Discrete Case  Bell System Technical Journal 57 (1978)所提出。

這些函數有些不同的意涵，當在解亥姆霍兹方程，透過在長球面坐標系做變數分離，使得各代表：

  and .

得到解為長球波函數與角球波函數的成積乘上. 這裡的變數c可定義為, with 為長球的橢圓截面的兩焦點的距離。

徑向波(The radial wave function)滿足線性常微分方程:

此本徵值在Sturm-Liouville 微分方程中已被固定，透過設定為有限函數，當 .

角波函數滿足下列微分方程：

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)，在角波函數中)。

當給定，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，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，函數將滿足以下線性微分方程：

此函數為球波函數。這個輔助方程式在StrattonJ. A. Stratton Spheroidal functions  Proceedings of the National Academy of Sciences (USA) 21, 51 (1935) 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.. M. Abramowitz and I. Stegun. Handbook of Mathematical Functions pp. 751-759  (Dover, New York, 1972)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C. Flammer. Spheroidal Wave Functions Stanford University Press, Stanford, CA, 1957。

一開始，球波函數是由C. Niven,C. Niven )On the conduction of heat in ellipsoids of revolution.  Philosophical transactions of the Royal Society of London, 171 p. 117 (1880)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,M. J. O. Strutt. Lamesche, Mathieusche and Verdandte Funktionen in Physik und Technik Ergebn. Math. u. Grenzeb, 1, pp. 199-323, 1932 Stratton et al.,J. A. Stratton, P. M. Morse, J. L. Chu, and F. J. Corbato. 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.等人的作品。

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,B. J. King, R. V. Baier, and S Hanish A Fortran computer program for calculating the prolate spheroidal radial functions of the first and second kind and their first derivatives.  (1970) Patz和Van Buren,B. J. Patz and A. L. Van Buren A Fortran computer program for calculating the prolate spheroidal angular functions of the first kind.  (1981) Baier與其團隊,R. V. Baier, A. L. Van Buren, S. Hanish, B. J. King - Spheroidal wave functions: their use and evaluation  The Journal of the Acoustical Society of America, 48, pp. 102–102 (1970) Zhang和Jin,S. Zhang and J. Jin. Computation of Special Functions, Wiley, New York, 1996 Thompson,W. J. Thomson Spheroidal Wave functions  Computing in Science & Engineering p. 84, May–June 1999、Falloon.P. E. Falloon Thesis on numerical computation of spheroidal functions  University of Western Australia, 2002 Van Buren和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 Mathemathics 60, pp. 589-599, 2002A. 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.。 

Flammer, Hunter,H. E. Hunter Tables of prolate spheroidal functions for m=0: Volume I.  (1965)H. E. Hunter 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 Tables of radial spheroidal wave functions, volume 1, prolate, m = 0 (1970)S. Hanish, R. V. Baier, A. L. Van Buren, and B. J. King Tables of radial spheroidal wave functions, volume 2, prolate, m = 1  (1970)S. Hanish, R. V. Baier, A. L. Van Buren, and B. J. King Tables of radial spheroidal wave functions, volume 3, prolate, m = 2  (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。

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"F. J. Simons, M. A. Wieczorek and F. A. Dahlen. Spatiospectral concentration on a sphere. SIAM Review, 2006,  (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F. J. Simons and Dahlen, F. A. Spherical Slepian functions and the polar gap in Geodesy. Geophysical Journal International, 2006, 以及宇宙學.F. A. Dahlen和F. J. Simons. Spectral estimation on a sphere in geophysics and cosmology. Geophysical Journal International, 2008, 

 參考文獻 

 外部連結 
 MathWorld Spheroidal Wave functions 
 MathWorld Prolate Spheroidal Wave Function 
 MathWorld Oblate Spheroidal Wave function 

Category:特殊函数
Category:小波分析