Jen l and hu c 1983 spheroidal wave functions of large frequency parameters c kf, the radiation fields of a metallic prolate spheroid excited by any circumferential slot ieee trans. 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. Solutions to a generalized spheroidal wave equation. Pdf prolate spheroidal wave functions associated with the. The current version does not implement the classic onedimensional prolate spheroidal wave functions pswf, which we plan to add in the future.
Flammer, spheroidal wave functions, 1957 are included as a special case of meixners more general functions. The sinc function in the sampling theory can be replaced by the prolate spheroidal wave functions pswfs, or slepian functions, since they have much better time localization than the sinc. Expression of generalized prolate functions for sets. Steerable principal components for spacefrequency localized. The magic of the prolate spheroidal functions in various setups. Introduction the prolate spheroidal wave functions pswfs originally appeared during the resolution of the helmholtz equation in the spheroidal coordinate system by variables separation 16.
Let denote the time truncation operator, such that if and only if is timelimited within. Publishers pdf, also known as version of record includes final page, issue. Software for computing the spheroidal wave functions using arbitrary precision arithmetic ross adelman, nail a. In order to be able to accurately detect these zeros in the presence of noise, the received signal is filtered using a prolate spheroidal wave function based model.
They are eigenfunctions of a timelimiting operation followed by a lowpassing operation. The discrete prolate spheroidal sequences dpsss are the discrete form of the prolate spheroidal wave functions pswfs 16 and can be used as the basis for the projection of the sampled signal. Despite their useful properties in many applications, gpsfs are often replaced by crude approximations. Spectral methods based on prolate spheroidal wave functions. 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. Pdf theory and computation of spheroidal wavefunctions. This site provides pdf files of the three articles published in the quarterly of applied mathematics that are the basis for matfcn and profcn.
Properties of the functions are derived and several applications to the representation of signals are made. P prolate spheroidal wave functions as an alternative to. Theory and computation of spheroidal wavefunctions. Jul 06, 2015 what is a prolate spheroid wave function and how does it apply to eegs and brain mapping. Spheroidal wave functions dover books on mathematics kindle edition by flammer, carson. The oblate spheroidal wave functions sciencedirect. It also provides an article submitted to arxiv describing. Mathieu and spheroidal wave functions fortran programs.
The prolate spheroidal wave functions, which are a special case of the spher. Spheroidal wave functions by carson flammer overdrive. Aline bonamiaand abderrazek karouib1 a f ed eration denispoisson, mapmoumr 6628, department of mathematics, university of orl eans, 45067 orl eans cedex 2, france. The tables were computed to fiveplace accuracy on the whirlwind digital computer, and automatically tabulated.
An example taken from atomic physics is the scattering of electrons and positrons by diatomic molecules. Spheroidal wave functions become elementary if gammanpi2. Derivation of prolate spheroidal wave functions there are several ways to generate the function set, each having its advantages and disadvantages in complexity and precision 3,7,18,24. We demonstrate the pros and cons over their polynomial counterparts, and put the emphasis on the construction of essential building blocks for efficient spectral algorithms. They have been largely studied and used after the seminal work of d. Timevariant channel estimation using discrete prolate spheroidal sequences thomas zemen, member, ieee, and christoph f. Much of the theory and numerical machinery for pswfs in one dimension is fairly complete see, for example, 21 and 15.
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. Eigenvalues of spheroidal wave functions and their branch. Spheroidal wave functions dover books on mathematics. Prolate radial spheroidal wave functions article pdf available in computer physics communications 302. Later it is discovered that they are essential for the description of electromagnetic wave. For the high dimensional version, see prol matlab version. Osa generalized prolate spheroidal wave functions for. Alternative expressions for calculating the oblate spheroidal radial functions of. This happens at the extreme kerr limit of black hole rotation, and concludes the present analysis of generalized spheroidal wave functions. The radiated acoustic pressures are obtained by means of an expansion of independent functions based on the prolate spheroidal coordinate system. Spheroidal wave function and its eigenvalues for complex size parameter benjamin e. 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. Kong abstract we provide a rapid and accurate method for calculating the prolate and oblate spheroidal wave functions pswfs and oswfs, smnc.
The systems must have a common origin of coordinates and a common symmetry axis of coordinate surfaces. Prolate spheroidal wave function how is prolate spheroidal. Computation of the discrete prolate spheroidal window. Prolate spheroidal wave functions and their generalisations, timefrequency concentration problem, bandlimited functions. Pdf prolate spheroidal wave functions, an introduction. Prolate spheroidal wave functionsprolate elements 151. Prolate spheroidal wave functions, fourier analysis and uncertainty ii. Discrete prolate spheroidal wave functions and interpolation. Prolate spheroidal wave functions, fourier analysis and uncertainty i by d. In mathematics, the prolate spheroidal wave functions pswf are a set of orthogonal bandlimited functions. A complete set of bandlimited functions is described which possesses the curious property of being orthogonal over a given finite interval as well as over. 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.
These are of great utility in disciplines such as geodesy 28 or cosmology. On the asymptotic expansion of the spheroidal wave function. Wedescribe an algorithm for the interpolation ofburst errors in discretetime signals that can be modelled as being bandlimited. Prolate spheroidal wave functions of order zero springerlink. Pdf reducing sampling error by prolate spheroidal wave. On the asymptotic expansion of the spheroidal wave. Niven, which lead to a helmholtz equation in spheroidal coordinates. 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. Improved detection scheme for chipless rfids using prolate. Software for computing the spheroidal wave functions using. Software for computing the spheroidal wave functions. Spectral decay of the sinc kernel operator and approximations by prolate spheroidal wave functions.
Prolate spheroidal wave functions, fourier analysis and uncertainty iii. Separation of variables of the wave equation in spheroidal coordinates results in a differential equation that is satis. Introduction clifford analysis offers both a generalization of complex analysis in the plane and a re. Bandlimited signal extrapolation using prolate spheroidal. The magic of the prolate spheroidal functions in various.
Prolate spheroidal wave functions pswfs are the eigenfunctions of the bandlimited operator in one dimension. Journal of classical analysis volume 11, number 1 2017, 121 doi. Approximate formulae for certain prolate spheroidal wave functions valid for large values of both order and bandlimit. Obtained by flammer spheroidal wave functions stanford university press. Section viii looks at what happens to the generalized spheroidal wave equation and its coulomb wavefunction solutions sec. The detection is based on finding the zeros in the derivative of the group delay of the received signal. Currently, only a matlab implementation is available.
Generalized prolate spheroidal functions, gpsf, pswf, truncated fourier transform, implementation. Prolate spheroidal wave functions, an introduction to the slepian series and its properties by ian c. Asymptotics of prolate spheroidal wave functions t. Prolate spheroidal wave functions, an introduction to the. Efficient recovery of smooth functions which are ssparse with respect to the basis. These charts show that the eigenvalues have many branch points in the complex planes of their arguments. Prolate spheroidal wave functions pswfs play an important role in various areas, from physics e. Prolate spheroidal wave functions, fourier analysis and uncertainty i. Accurate calculation of oblate spheroidal wave functions arxiv. Some relations between the spheroidal and spherical wave.
The magic of the prolate spheroidal functions in various setups f. Explicit upper bounds on the eigenvalues associated with. He is best known for his work with algebraic coding theory, probability theory, and distributed source coding. On the theory ofspheroidal wave functions of order zero. Download it once and read it on your kindle device, pc, phones or tablets. Additional subjects include recurrence relations of whittaker type, asymptotic expansions for large values of c, and vector wave functions. He was colleagues with claude shannon and richard hamming at bell labs. Pollak manuscript received august 1, 1960 a complete set of bandlimited functions is.
A new formulation based on a prolate spheroidal coordinate was proposed to describe acoustic pressure in the acoustic field. Monographs tying together many aspects of the theory of spheroidal wave functions were written by strutt, stratton et al. Miranian department of mathematics, university of california, berkeley ca, 94720 abstract the singular functions for the problem of recovering a time limited function from its fourier transform in a certain. Prolate spheroidal wave functions and applications. Spheroidalradialfactor nu,mu,gamma 1 formula primary definition 1 formula spheroidalradialfactor nu, mu, gamma 1 formula. Spheroidal wave functions flammer pdf spheroidal wave functions of flammer c. Theory and computation of the spheroidal wave functions. Pdf high order domain decomposition methods using a basis of. Abstract generalized prolate spheroidal functions gpsf are the eigen functionsofthetruncatedfouriertransform, restrictedtoddimensional balls in the spatial domain and frequency domain. Then the acoustic pressure in the field can be reconstructed by using some measured acoustic pressure signals in the field. Generalized prolate spheroidal wave functions for offset. A set of tables of spheroidal wave functions designed to simplify the computation of acoustic and electromagnetic scattering from spheroids. Prolate spheroidal wave functions, an introduction to the slepian series and its properties author links open overlay panel ian c. We extend the theory of pswfs for the finiteextension fractional fourier transform, the finiteextension linear canonical transform, and the finiteextension offset linear canonical transform.
Prolate spheroidal wave functions, quadrature, and. I have been working on algorithms for spheroidal wave functions for 50 years. The algorithm correctly restores a mutilated signal that is indeed bandlimited. Prolate spheroidal wave functions pswf in development. Prolate spheroidal wave functions and discrete prolate spheroidal sequences the continuous time problem of maximally concentrating a timelimited function to a limited bandwidth was considered by slepian, pollak, and landau and they found that the prolate spheroidal wave functions pswf are maximally concentrated in this regard 811. Prolate spheroidal wave function as exact solution of the. Prolate spheroidal wave functions of order zero download prolate spheroidal wave functions of order zero ebook pdf or read online books in pdf, epub, and mobi format. Prolate spheroidal wave functions 811 we next show that. Manuscript received august 1, 1960 a complete set of bandlimited functions is described which possesses the curious property f being orthogonal over a given finite interval as well as over 00, 00. Prolate spheroidal wave functions, an introduction to the slepian series and its properties. Prolate spheroidal wave functions associated with the quaternionic fourier transform article pdf available in mathematical methods in the applied sciences may 2017 with 98 reads. Spectral decay of the sinc kernel operator and approximations.
Prolate spheroidal wave functions pswfs are the natural basis for representing bandlimited functions on the interval. Read computational techniques for prolate spheroidal wave functions in signal processing, journal of computational methods in sciences and engineering on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at. Asymptotic expansions of angular prolate spheroidal wave functions for large values of. 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. These finite transforms are more flexible than the fift. Slepian june 30, 1923 november 29, 2007 was an american mathematician. Pswf prolate spheroidal wave function acronymfinder. The transition matrix for acoustic and elastic wave scatte ring in prolate spheroidal coordinates roger h. What is a prolate spheroid wave function and how does it apply to eegs and brain mapping. The spheroidal wave functions, which are the solutions to the helmholtz. Prolate spheroidal wave functions, fourier analysis and uncertainty i slepian 1961 bell system technical journal wiley online library skip to article content.
Recent investigations of electromagnetic problems in regions bounded by spheroids have led to an mcreased interest in the spheroidal wave functions. Prolate spheroidal wave functions, fourier analysis and. Flammer, spheroidal wave functions, stanford university press. 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. As such, they play an important role in signal processing, fourier analysis, and approximation theory. Generalized prolate functions and another isomonodromy problem. Rokhlin, highfrequency asymptotic expansions for certain prolate spheroidal wave functions, the journal of fourier analysis and applications, 9 2003. The applicability ranges of the relations obtained are discussed. A novel, highly sensitive scheme to detect the resonance peaks in the spectrum of chipless rfid signals is presented. Prolate spheroidal functions mathieu and spheroidal wave. Click download or read online button to prolate spheroidal wave functions of order zero book pdf. On generalized prolate spheroidal functions preliminary. This implementation does not scale well at this point due to inefficient use of matlabs eigendecomposition, which will be replaced in future versions. While historically the numerical evaluation of pswfs presented serious difficulties, the.
An introduction discusses the mathematical properties of the functions and describes some of their applications. The transition matrix for acoustic and elastic wave. An op1q algorithm for the numerical evaluation of the prolate. First we consider the case where a parameter becomes large and positive, and provide rigorous results for the prolate spheroidal wave functions and their corresponding eigenvalues, which are similar to those given in the. The output of oblfcn includes diagnostic files one for radial functions and one for.
Eigenvalues of spheroidal wave functions are calculated as functions of complex propagation constants. The dpsw is a scalar multiple of the zeroorder dpss for a given n and w, and is given by an eigenvector associated with the dominant eigenvector of the matrix tn,w. Download pdf prolate spheroidal wave functions of order. Prolate spheroidal wave functions, fourier analysis and uncertainty iv. This code is under development and it is still being tested. Kaiser windows are in fact generated by sampling this i0sinh approximation. Spheroidaleigenvaluenu,mu,gamma 11 formulas primary definition 1 formula specific values 6 formulas. 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. Compressed sensing with preconditioning for sparse recovery with. On the use of discrete prolate spheroidal windows for. The theory developed in the preceding paper is applied to a number of questions about timelimited and bandlimited signals. Discrete prolate spheroidal sequences and wave functions.
An op1q algorithm for the numerical evaluation of the. We explore a number of issues related to this problem, including the effective computation of the corresponding slepzan functions for. Use features like bookmarks, note taking and highlighting while reading spheroidal wave functions dover books on mathematics. Discrete prolate spheroidal slepian sequences matlab dpss. The solution of 1 is the prolate radial spheroidal function, which is often encountered in scattering problems describing the longrangepart of the system wave function.
Dunster san diego state university san diego, ca 92182 usa abstract. Uniform asymptotic approximations are obtained for the prolate spheroidal functions ps n mx. Introduction the oblate spheroidal wave functions arise as the eigenfunctions of such boundary value problems as the diffraction of sound and electro magnetic waves by circular disks and apertures, and the oscillations of spheroidal cavities. Computational techniques for prolate spheroidal wave. Contents 1 introduction 102 2 prolate spheroidal wave functions 105. Within the last ten years, computer processing speeds have climbed to the point where generating the function set is not an issue when it comes to their study. Originally, the spheroidal wave functions were introduced by c. 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. This prediction is effectively relevant since the pswfs are deeply studied in mathematical physics and are classically defined as a series of normalized legendre polynomials which are well known as special functions of mathematics.
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