Software for computing the spheroidal wave functions. They also find considerable application in other areas such as signal processing and. By a dielectric coated spheroidal particle elsayed esam m. Theory and computation of the spheroidal wave functions. On generalized prolate spheroidal functions preliminary. Chromatic series with prolate spheroidal wave functions. Stratton 1935, chu and stratton 1941, and rhodes 1970 define the spheroidal functions as those solutions of the differential equation. The dimension of the space of essentially time and bandlimited signals. An op1q algorithm for the numerical evaluation of the prolate. 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 cimpa school on real and complex analysis with applications, buea cameroun, 114 may 2011. Prolate spheroidal wave functions, fourier analysis and.
The tables were computed to fiveplace accuracy on the whirlwind digital computer, and automatically tabulated. We then discuss the important special case for the angular functions of the second kind. Prolate spheroidal wave functions, an introduction to the slepian series and its. They represented the solution in terms of prolate spheroidal wave functions through the use of separation of variables of the scalar. Mathematics of computation volume 79, number 270, april 2010, pages 807827 s 0025571809022686 article electronically published on september 17, 2009. Prolate spheroidal wave functions pswfs play an important role in various areas, from physics e. Spheroidaleigenvaluenu,mu,gamma 11 formulas primary definition 1 formula specific values 6 formulas. Prolate spheroidal wave functions, an introduction to the slepian series and its properties. Jul 06, 2015 what is a prolate spheroid wave function and how does it apply to eegs and brain mapping.
Wedescribe an algorithm for the interpolation ofburst errors in discretetime signals that can be modelled as being bandlimited. Flammer summarizes the work up to 1957 19 and documents the. 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. These are of great utility in disciplines such as geodesy or cosmology. Prolate spheroidal wave functions, fourier analysis and uncertainty i slepian 1961 bell system technical journal wiley online library skip to article content. Pswf prolate spheroidal wave function acronymfinder. Prolate spheroidal wave functions, fourier analysis and uncertainty ii. Li et al 5, solved the problem of the electromagnetic radiation from a prolate spheroidal antenna enclosed in a confocal radome spheroidal dielectric covers. Separation of variables of the wave equation in spheroidal coordinates results in a differential equation that is satis. 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.
Download pdf spheroidal wave functions free online new. An introduction discusses the mathematical properties of the functions and describes some of their applications. This paper discusses the calculation of prolate spheroidal wave functions for integral mode numbers and real spheroidal parameter. Spheroidal wave functions download spheroidal wave functions ebook pdf or read online books in pdf, epub, and mobi format. 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. In particular, the theory of spheroidal wave functions in the shortwavelength limit has recently been signi. Pdf theory and computation of spheroidal wavefunctions. Two dimensional prolate spheroidal wave functions for mri. Asymptotic expansions of angular prolate spheroidal wave functions for large values of. The spheroidal library includes routines for computing the spheroidal wave functions, and is located in the spheroidal directory. Spheroidal wave functions dover books on mathematics. Prolate spheroidal wave functions, an introduction to the. Flammer, spheroidal wave functions, 1957 are included as a special case of meixners more general functions. Use features like bookmarks, note taking and highlighting while reading spheroidal wave functions dover books on mathematics.
Spheroidal wave functions by carson flammer overdrive. We explore a number of issues related to this problem, including the effective computation of the corresponding slepzan functions for. Prolate spheroidal wave functions whose domain is a portion of the surface of the unit sphere are more generally called slepian functions see also spectral concentration problem. Discrete prolate spheroidal sequences and wave functions. It is assumed that negativeenergymass particles in dirac sea be considered in physical interactions, and that physical space consist of 4dimensional complex space, in another words, each dimension has real and imaginary parts. It appears then logic to investigate the existence of a schrodinger equation which admits the pswfs as exact analytical solutions.
Prolate spheroidal wave function as exact solution of the. A set of tables of spheroidal wave functions designed to simplify the computation of acoustic and electromagnetic scattering from spheroids. Prolate spheroidal wave functions, fourier analysis and uncertainty iii. Approximation by other special function classes 42a38. Request pdf two dimensional prolate spheroidal wave functions for mri the tradeoff between spatial and temporal resolution is often used to increase data acquisition speed for dynamic mr imaging. In this paper we examine the merits of using prolate spheroidal wave functions as basis functions in pseudospectral solution of hyperbolic partial differential. Under these conditions one may ask whether there is a schrodinger equation whose exact eigensolutions are the prolate spheroidal wave functions. 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. 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. I have been working on algorithms for spheroidal wave functions for 50 years. Publishers pdf, also known as version of record includes final page, issue. Solutions to this equation constitute an essential element in the numerical calculation of the diffraction, radiation, and scattering of acoustic and electromagnetic waves by spheroids. The effect ofwindowing can also be described conveniently in terms ofthese sequences and functions.
Scattering of a focused gaussian beam 1171 where h, and d mn tare constants depend on the incident beam. The functions have applications in electromagnetism, acoustics, quantum mechanics and finite fourier analysis. Spheroidal wave functions flammer pdf spheroidal wave functions of flammer c. Prolate spheroidal wave functions slepian functions are special functions that are most local. Additional subjects include recurrence relations of whittaker type, asymptotic expansions for large values of c, and vector wave functions. It also provides an article submitted to arxiv describing. Explicit upper bounds on the eigenvalues associated with. What is a prolate spheroid wave function and how does it apply to eegs and brain mapping. The magic of the prolate spheroidal functions in various setups. On the asymptotic expansion of the spheroidal wave. Addressed to applied mathematicians, mathematical physicists, and mathematical engineers, it presents tables that provide a convenient means for handling wave problems in spheroidal coordinates. Prolate spheroidal wave functions pswfs are the natural basis for representing ban dlimited functions on the interval. The spheroidal wave functions, which are the solutions to the helmholtz. Spectral methods based on prolate spheroidal wave functions.
Click download or read online button to spheroidal wave functions book pdf for free now. The algorithm correctly restores a mutilated signal that is indeed bandlimited. This repository contains four libraries from the matlab central file exchange. Much of the theory and numerical machinery for pswfs in one dimension is fairly complete see, for example, 21 and 15.
Journal of classical analysis volume 11, number 1 2017, 121 doi. Software for computing the spheroidal wave functions using arbitrary precision arithmetic ross adelman, nail a. The systems must have a common origin of coordinates and a common symmetry axis of coordinate surfaces. This paper relies heavily on the theory of spheroidal wave functions as found elsewhere 10,11. Click here for the pdf file of the article accurate calculation of prolate spheroidal radial functions of the first kind and their first derivatives, quart. Mathieu and spheroidal wave functions fortran programs. The spheroidal wavefunctions of flammer 1957 spheroidal wave functions are included as a special case of meixners more general functions. Prolate spheroidal wave functions computer science yale.
Download it once and read it on your kindle device, pc, phones or tablets. Spheroidaleigenvaluenu,mu,gamma 11 formulas primary definition 1 formula specific values 6 formulas general characteristics 4 formulas spheroidaleigenvalue nu, mu, gamma 11 formulas spheroidaleigenvalue. The resulting method does not provide a numerically stable mechanism for evaluating the functions ps npx. Discrete prolate spheroidal wave functions and interpolation. A wavelet in the form of the first odd prolate spheroidal wave function is proposed for the wavelet transform of a nonuniform 2d image and the formation of clusters of wavelet coefficients in it. In this paper we consider discretetime, real or complexvalued signals stskkz. Spheroidal wave functions become elementary if gammanpi2. Calculation of spheroidal wave functions sciencedirect. Software for computing the spheroidal wave functions using. An op1q algorithm for the numerical evaluation of the. Approximate formulae for certain prolate spheroidal wave functions valid for large values of both order and bandlimit. 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. On the asymptotic expansion of the spheroidal wave function. Rokhlin, highfrequency asymptotic expansions for certain prolate spheroidal wave functions, the journal of fourier analysis and applications, 9 2003.
Methods for calculating the wavelet function, clustering the field of wavelet coefficients, and constructing their corresponding optimal rectangular windows in an image are described. New relations are found between the spheroidal and spherical wave functions, as well as between the spheroidal functions related to different spheroidal coordinate systems. One of the principal reasons for the importance of pswfs is that they are a. The applicability ranges of the relations obtained are discussed. They are called oblate spheroidal wave functions if oblate spheroidal coordinates are used and prolate spheroidal wave functions if prolate. The objective is the use of large values of mode number and spheroidal parameter.
Some relations between the spheroidal and spherical wave. Eigenvalues of spheroidal wave functions and their branch. Bandlimited signal extrapolation using prolate spheroidal. These charts show that the eigenvalues have many branch points in the complex planes of their arguments.
On the evaluation of prolate spheroidal wave functions and. Waveletodd prolate spheroidal wave functions in the. Eigenvalues of spheroidal wave functions are calculated as functions of complex propagation constants. However, the threeterm recurrence relations satis ed by the legendre coe cients are then used to construct the expansions of the prolate spheroidal wave functions.