As such, they play an important role in signal processing, fourier analysis, and approximation theory. The objective is the use of large values of mode number and spheroidal parameter. Spatial smoothing in fmri using prolate spheroidal wave. The matlab versions are direct machine conversions of the fortran sources via. Obtained by flammer spheroidal wave functions stanford university press. Volkmer department of mathematical sciences, university of wisconsinmilwaukee, milwaukee, wisconsin acknowledgements. Topics include separation of the scalar wave equation in spheroidal coordinates, angle and radial functions, integral representations and relations, and expansions in spherical bessel function products. In this paper we show that using discrete prolate spheroidal sequences, rather than sine functions, it is possible to derive a sampling and reconstruction method which is similar to cs.
Assuming nonuniform sampling our procedure can be connected with compressive sensing without complex reconstruction methods. Jin, which accompany their book computation of special functions wiley, 1996. Spheroidal wave functions are solutions of the helmholtz equation that are found by writing the. These functions 8 mnc, 1, rmnc, result from the solution of the scalar wave equation by separation. Prolate spheroidal wave function as exact solution of the.
The method is practical, highly accurate, and applicable to a wide range of the parameters m, n and c. In mathematics, the prolate spheroidal wave functions are a set of functions derived by timelimiting and lowpassing, and a second timelimit operation. We explore a number of issues related to this problem, including the effective computation of the corresponding slepzan functions for. Computation of special functions file exchange matlab. This incarnation of the prolates is studied thoroughly in the monographs of flammer 10, meixner and sch afke 17, stratton et al 21 and morse and feshbach 16 among others. Spheroidal coordinates and wave functions dyadic greens functions in spheroidal systems em scattering by a conducting spheroid em scattering by a coated dielectric spheroid spheroid antennas sar distributions. Abstract not available bibtex entry for this abstract preferred format for this abstract see preferences find. Gumerov, and ramani duraiswami abstract the spheroidal wave functions, which are the solutions to the helmholtz equation in spheroidal coordinates, are notoriously di cult to compute. Prolate radial spheroidal wave functions sciencedirect. The prolate spheroidal wave function filter the prolate spheroidal wave function pswf filter is the function, with compact support on a fixed set of kspace, which maximizes the. A computational method to calculate the exact solution for. Bandlimited signal extrapolation using prolate spheroidal.
Spheroidal wave functions by carson flammer overdrive. The applicability ranges of the relations obtained are discussed. The magic of the prolate spheroidal functions in various setups. This paper discusses the calculation of prolate spheroidal wave functions for integral mode numbers and real spheroidal parameter. Flammer, spheroidal wave functions, 1957 are included as a special case of meixners more general functions. The orthogonal properties of linear prolate spheroidal wave functions pswfs are exploited to form an orthogonal basis set needed for synthesis. Use features like bookmarks, note taking and highlighting while reading spheroidal wave functions dover books on mathematics. Prolate spheroidal wave functions and their properties computation of the pswfs by flammers method uniform estimates of the pswfs and their derivatives applications of the pswfs prolate spheroidal wave functions from pde point of view pswfs as eigenfunctions of a di erential and an integral operator some properties of the pswfs spheroidal.
What links here related changes upload file special pages permanent link page. Prolate spheroidal wave functions pswfs play an important role in various areas, from physics e. Software for computing the spheroidal wave functions using. Yingle li, jin li, mingjun wang, qunfeng dong keywords. A significant step in the process is the higher order piecewise polynomial approximation of the overlap integral required for obtaining the expansion coefficients accurately with very high precision. Jul 06, 2015 what is a prolate spheroid wave function and how does it apply to eegs and brain mapping. They are called oblate spheroidal wave functions if oblate spheroidal coordinates are used and prolate spheroidal wave functions if prolate. Flammer, spheroidal wave functions, stanford university press. The flagship monograph addressing the spheroidal wave function andits pertinence to computational electromagnetics spheroidal wave functions in electromagnetic theorypresents in detail the theory of spheroidal wave functions, itsapplications to the analysis of electromagnetic fields in variousspheroidal structures, and provides comprehensive programming codesfor those computations. Recurrence relations for prolate spheroidal wave functions. Prolate spheroidal wave functions pswfs are the eigenfunctions of the bandlimited operator in one dimension. Accurate calculation of oblate spheroidal wave functions arxiv. The prolate spheroidal wavefunctions pswfs or prolates have long been studied in the context of solving the wave equation in prolate spheroidal coordinates.
Niven, which lead to a helmholtz equation in spheroidal coordinates. The operator turns out to be linear, bounded and selfadjoint. Stratton 1935, chu and stratton 1941, and rhodes 1970 define the spheroidal functions as those solutions of the differential equation. Software for computing the spheroidal wave functions. Prolate spheroidal wave functions and their properties computation of the pswfs by flammer s method uniform estimates of the pswfs and their derivatives applications of the pswfs wkb method for the pswfs uniform bounds of the pswfs and their derivatives exponential decay of the eigenvalues associated with the pswfs. Eigenvalues of spheroidal wave functions are calculated as functions of complex propagation constants. International journal of imaging systems and technology, 1262. Recently, pswfs have been becoming increasingly popular in various areas in which such functions occur this includes physics e. Spheroidal wave functions by carson flammer waterstones.
An op1q algorithm for the numerical evaluation of the prolate. It is therefore reasonable to be interested in expressing the solution of the schrodinger equation in terms of prolate spheroidal wave functions. Addressed to applied mathematicians, mathematical physicists, and mathematical engineers, it discusses separation of the scalar wave equation in spheroidal coordinates, angle and radial functions, integral representations and relationism, and expansio. One of the principal reasons for the importance of pswfs is that they are a. What links here related changes upload file special pages permanent link page information wikidata item cite this page. Explicit upper bounds on the eigenvalues associated with. Software for computing the spheroidal wave functions using arbitrary precision arithmetic ross adelman, nail a. Computation of special functions file exchange matlab central. For many purposes the calculation of the eigenvalues and the regular angular function can be regarded as completely solved. The singular functions for the problem of recovering a time limited function from its fourier transform in a certain band of frequencies are given, in the simplest case, by the prolate spheroidal wave functions. Apr 14, 2005 this collection of special mathematical functions originated in fortran programs by s.
Download spheroidal wave functions ebook pdf or read online books. In applied mathematics, oblate spheroidal wave functions are involved in the solution of the. The output of oblfcn includes diagnostic files one for radial functions and one for. New relations are found between the spheroidal and spherical wave. Theory and computation of the spheroidal wave functions.
Spheroidal wave functions flammer pdf spheroidal wave functions of flammer c. Intended to facilitate the use and calculation of spheroidal wave functions, this applicationsoriented text features a detailed and unified account of the properties of these functions. Recent investigations of electromagnetic problems in regions bounded by spheroids have led to an mcreased interest in the spheroidal wave functions. Monographs tying together many aspects of the theory of spheroidal wave functions were written by strutt, stratton et al. Abstract a simple, efficient and reliable method to extrapolate bandlimited signals varying from lower to higher frequencies is proposed. In the next few sections we follow the notation of flammer 3 and slepian 18,19 where pswfs. Spheroidaleigenvaluenu,mu,gamma 11 formulas primary definition 1 formula specific values 6 formulas general characteristics 4 formulas spheroidaleigenvalue nu, mu, gamma 11 formulas spheroidaleigenvalue. Additional subjects include recurrence relations of whittaker type, asymptotic expansions for large values of c, and vector wave functions. The property of rayleigh scattering for an anisotropic dielectric ellipsoid authors. Volkmer department of mathematical sciences, university of wisconsinmilwaukee, milwaukee, wisconsin. Optimal data acquisition in fmri using prolate spheroidal wave functions. Calculation of spheroidal wave functions sciencedirect. Cdmpeanu prolate radial spheroidal wave functions fig. Download it once and read it on your kindle device, pc, phones or tablets.
Similarly, let denote an ideal lowpass filtering operator, such that iff x is bandlimited within. Wave function prolate radial function spheroidal particle spheroidal wave function. Monographs tying together many aspects of the theory of spheroidal wave functions were written by strutt, 10 stratton et al. Flammer summarizes the work up to 1957 19 and documents the. Originally, the spheroidal wave functions were introduced by c. The functions have applications in electromagnetism, acoustics, quantum mechanics and finite fourier analysis. Application of nonorthogonal bases in the theory of light. Pswf prolate spheroidal wave function acronymfinder. Contents 1 introduction 102 2 prolate spheroidal wave functions 105. A set of tables of spheroidal wave functions designed to simplify the computation of acoustic and electromagnetic scattering from spheroids. Spheroidal wave functions dover books on mathematics. Some relations between the spheroidal and spherical wave. The spheroidal wavefunctions of flammer 1957 spheroidal wave functions are included as a special case of meixners more general functions. Buy spheroidal wave functions by carson flammer from waterstones today.
Prolate spheroidal wave functions and applications. The prolate spheroidal wave functions, which are a special case of the spher. On the asymptotic expansion of the spheroidal wave function and. The magic of the prolate spheroidal functions in various. Journal of classical analysis volume 11, number 1 2017, 121 doi. On the evaluation of prolate spheroidal wave functions and.
An introduction discusses the mathematical properties of the functions and describes some of. The matlab versions are direct machine conversions of the fortran sources via an automatic translation program. The tables were computed to fiveplace accuracy on the whirlwind digital computer, and automatically tabulated. Over 10 million scientific documents at your fingertips. Addressed to applied mathematicians, mathematical physicists, and mathematical engineers, it presents tables that provide a convenient means for handling wave. Spheroidal wave functions become elementary if gammanpi2. This collection of special mathematical functions originated in fortran programs by s. Prolate spheroidal wave functions, an introduction to the slepian series and its properties. This applicationsoriented text features a detailed and unified account of the use and calculation of spheroidal wave functions. Mar 11, 2016 new relations are found between the spheroidal and spherical wave functions, as well as between the spheroidal functions related to different spheroidal coordinate systems. Soleski,a new friendly method of computing prolate spheroidal wave func tions and wavelets, appl. Numerical test calculations have demonstrated a high. Figure 1 omitted figure 2 omitted figure 3 omitted figure 4 omitted figure 5 omitted 5 conclusions we have studied properties of prolate spheroidal wave functions pswfs that are bandlimited to 12, 12 and approximately timelimited to t, t for a fixed t 0.
Spheroidal wave functions are solutions of the helmholtz equation that are found by writing the equation in spheroidal coordinates and applying the technique of separation of variables, just like the use of spherical coordinates lead to spherical harmonics. Prolate spheroidal wave functions, fourier analysis and uncertainty iv. An introduction discusses the mathematical properties of the functions and describes some of their applications. R okhlin, highfrequency asymptotic expansions for certain prolate spheroidal wave functions, j. Pdf theory and computation of spheroidal wavefunctions. Approximate formulae for certain prolate spheroidal wave functions valid for large values of both order and bandlimit. Prolate spheroidal wave functions, fourier analysis and uncertainty i. Alternative expressions for calculating the oblate spheroidal radial functions of both. These charts show that the eigenvalues have many branch points in the complex planes of their arguments. Download pdf spheroidal wave functions free online new. The theory of light scattering by single particles and their ensembles has important applications in various areas of science and technology, e. Prolate spheroidal wave functions, an introduction to the. New relations are found between the spheroidal and spherical wave functions, as well as between the spheroidal functions related to different spheroidal coordinate systems.
Addressed to applied mathematicians, mathematical physicists, and mathematical engineers, it presents tables that provide a convenient means for handling wave problems in spheroidal coordinates. The paper shows the utility of the matrix method in the calculation of spheroidal eigenvalues and wave functions. Tables of numerical values of oblate spheroidal wave functions are given in flammer, hanish et al. Spheroidal wave functions dover books on mathematics kindle edition by flammer, carson. Prolate spheroidal wave functions and their generalisations, timefrequency concentration problem, bandlimited functions. The systems must have a common origin of coordinates and a common symmetry axis of coordinate surfaces. Sep 22, 2005 intended to facilitate the use and calculation of spheroidal wave functions, this applicationsoriented text features a detailed and unified account of the properties of these functions. Eigenvalues of spheroidal wave functions and their branch. Let denote the time truncation operator, such that iff x is timelimited within. Prolate spheroidal wave function how is prolate spheroidal.
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