翻訳と辞書 Words near each other ・ Prolactin-releasing peptide ・ Prolactin-releasing peptide receptor ・ Prolactinoma ・ Prolacuna ・ Prolagus ・ Prolagus imperialis ・ Prolagus oeningensis ・ Prolagus osmolskae ・ Prolais ・ Prolamin ・ Prolapse ・ Prolapse (band) ・ Prolasius ・ Prolasius advenus ・ Prolate spheroidal coordinates ・ Prolate spheroidal wave function ・ Prolate trochoidal mass spectrometer ・ Prolates ・ Prolation ・ Prolation canon ・ Prolative case ・ Prolaz ・ Prole ・ Prole, Iowa ・ Proleb ・ Prolec GE ・ Prolecanitaceae ・ Prolecanitida ・ Prolecithophora ・ Prolecon Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Prolate spheroidal wave function ： ウィキペディア英語版 Prolate spheroidal wave function In mathematics, the prolate spheroidal wave functions are a set of functions derived by timelimiting and lowpassing, and a second timelimit operation. Let $Q\_T$ denote the time truncation operator, such that $x=Q\_T\; x$ if and only if (iff) x is timelimited within $()$. Similarly, let $P\_W$ denote an ideal low-pass filtering operator, such that $x=P\_W\; x$ iff x is bandlimited within $()$. The operator $Q\_T\; P\_W\; Q\_T$ turns out to be linear, bounded and self-adjoint. For $n=1,2,\backslash ldots$ we denote with $\backslash psi\_n$ the n-th eigenfunction, defined as : $\backslash \; Q\_T\; P\_W\; Q\_T\; \backslash psi\_n=\backslash lambda\_n\backslash psi\_n,$ where $\backslash \_n$ are the associated eigenvalues. The timelimited functions $\backslash \_n$ are the Prolate Spheroidal Wave Functions (PSWFs). Pioneering work in this area was performed by 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)〕 These functions are also encountered in a different context. When solving the Helmholtz equation, $\backslash Delta\; \backslash Phi\; +\; k^2\; \backslash Phi=0$, by the method of separation of variables in prolate spheroidal coordinates, $(\backslash xi,\backslash eta,\backslash phi)$, with: :$\backslash \; x=(d/2)\; \backslash xi\; \backslash eta,$ :$\backslash \; y=(d/2)\; \backslash sqrt\; \backslash cos\; \backslash phi,$ :$\backslash \; z=(d/2)\; \backslash sqrt\; \backslash sin\; \backslash phi,$ :$\backslash \; \backslash xi>=1$ and . the solution $\backslash Phi(\backslash xi,\backslash eta,\backslash phi)$ can be written as the product of a radial spheroidal wave function $R\_(c,\backslash xi)$ and an angular spheroidal wave function $S\_(c,\backslash eta)$ by $e^$. Here $c=kd/2$, with $d$ being the interfocal distance of the elliptical cross section of the prolate spheroid. The radial wave function $R\_(c,\backslash xi)$ satisfies the linear ordinary differential equation: :$\backslash \; (\backslash xi^2\; -1)\; \backslash frac\; +\; 2\backslash xi\; \backslash frac\; -\backslash left(\backslash lambda\_(c)\; -c^2\; \backslash xi^2\; +\backslash frac\backslash right)\; =\; 0$ The eigenvalue $\backslash lambda\_(c)$ of this Sturm-Liouville differential equation is fixed by the requirement that $$ must be finite for $|\backslash xi|\; \backslash to\; 1\_+$. The angular wave function satisfies the differential equation: :$\backslash \; (\backslash eta^2\; -1)\; \backslash frac\; +\; 2\backslash eta\; \backslash frac\; -\backslash left(\backslash lambda\_(c)\; -c^2\; \backslash eta^2\; +\backslash frac\backslash right)\; =\; 0$ 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, $\backslash xi>=1$) in the angular wave function ). For $c=0$ these two 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. Let us note that if one writes $S\_(c,\backslash eta)=(1-\backslash eta^2)^\; Y\_(c,\backslash eta)$, the function $Y\_(c,\backslash eta)$ satisfies the following linear ordinary differential equation: $\backslash \; (1-\backslash eta^2)\; \backslash frac\; -2\; (m+1)\; \backslash eta\; \backslash frac\; +\backslash left(c^2\; \backslash eta^2\; +m(m+1)-\backslash lambda\_(c)\backslash right)\; =\; 0,$ which is known as the spheroidal wave equation. This auxiliary equation is used for instance by Stratton〔J. A. Stratton ''(Spheroidal functions )'' Proceedings of the National Academy of Sciences (USA) 21, 51 (1935)〕 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.〔. M. Abramowitz and I. Stegun. ''Handbook of Mathematical Functions'' (pp. 751-759 ) (Dover, New York, 1972)〕 Abramowitz and Stegun (and the present article) follow the notation of Flammer.〔C. Flammer. ''Spheroidal Wave Functions'' Stanford University Press, Stanford, CA, 1957〕 Originally, the spheroidal wave functions were introduced by 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)〕 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 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.〔 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 ''(A Fortran computer program for calculating the prolate spheroidal radial functions of the first and second kind and their first derivatives. )'' (1970)〕 Patz and 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 et al.,〔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 and 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〕 and Falloon.〔P. E. Falloon (Thesis on numerical computation of spheroidal functions ) University of Western Australia, 2002〕 Van Buren and Boisvert〔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〕〔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〕 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. Tables of numerical values of spheroidal wave functions are given in 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〕 The Digital Library of Mathematical Functions http://dlmf.nist.gov provided by NIST is an excellent resource for spheroidal wave functions. 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, 2006, 〕 (see also Spectral concentration problem). These are of great utility in disciplines such as geodesy〔F. J. Simons and Dahlen, F. A. ''Spherical Slepian functions and the polar gap in Geodesy''. Geophysical Journal International, 2006, 〕 or cosmology.〔F. A. Dahlen and F. J. Simons. ''Spectral estimation on a sphere in geophysics and cosmology''. Geophysical Journal International, 2008, 〕 == References == 〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Prolate spheroidal wave function」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.