Kong abstract we provide a rapid and accurate method for calculating the prolate and oblate spheroidal wave functions pswfs and oswfs, smnc. Theory and computation of the spheroidal wave functions. This solution is a generalization of spheroidal wave function into the series form ofp n x. Pdf the prolate spheroidal wave functions pswfs are used in. Having the shape of a spheroid generated by rotating an ellipse about its longer axis. Asymptotics of prolate spheroidal wave functions t. Mathematical tools for bandlimited approximation applied mathematical sciences osipov, andrei, rokhlin, vladimir, xiao, hong on. It is therefore reasonable to be interested in expressing the solution of the schrodinger equation in terms of prolate spheroidal wave functions. Bandlimited signal extrapolation using prolate spheroidal. Prolate spheroidal wave functions, fourier analysis and uncertainty iv. Prolate spheroidal wave functions associated with the quaternionic fourier transform article pdf available in mathematical methods in the applied sciences may 2017 with 105 reads. Volkmer department of mathematical sciences, university of wisconsinmilwaukee, milwaukee, wisconsin acknowledgements.
Prolate spheroidal wave functions and applications. Journal of classical analysis volume 11, number 1 2017, 121 doi. Martin lindquist 1255 amsterdam ave, 10th floor, mc 4409. Prolate spheroid synonyms, prolate spheroid pronunciation, prolate spheroid translation, english dictionary definition of prolate spheroid. Discrete prolate spheroidal slepian sequences matlab dpss.
Prolate spheroidal wave functions pswfs are important functions in information. Prolate spheroidal wave functions, fourier analysis and uncertainty i. The twodimensional prolate spheroidal wave function 2dpswf method was previously introduced as an efficient method for tradingoff between spatial and temporal resolution in magnetic resonance imaging mri, with minimal penalty due to truncation and partial volume effects. Eigenvalues of spheroidal wave functions and their branch. Prolate spheroidal wave functions 811 we next show that. Product formulas for angular spheroidal wave functions on 0, n\ and for radial spheroidal wave functions on 0, oo are presented, which generalize results for the ultraspherical polynomials and functions as well as for the mathieu functions. Prolate spheroidal wave functions, fourier analysis, and. Use features like bookmarks, note taking and highlighting while reading prolate spheroidal wave functions of order. Convergent and divergent series, solutions of the prolate. Prolate spheroidal wave functions pswfs are known to be useful for analyzing the properties of the finiteextension fourier transform fift. Spheroidal wave function and its eigenvalues for complex size parameter benjamin e. We explore a number of issues related to this problem, including the effective computation of the corresponding slepzan functions for.
We extend the theory of pswfs for the finiteextension fractional fourier transform, the finiteextension linear canonical transform, and the finiteextension offset linear canonical transform. Wager2 1 department of statistics, columbia university, new york, ny, 10027 2 department of psychology, columbia university, new york, ny, 10027 address. Dunster san diego state university san diego, ca 92182 usa abstract. Jul 06, 2015 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. The operator turns out to be linear, bounded and selfadjoint. Much of the theory and numerical machinery for pswfs in one dimension is fairly complete see, for example, 21 and 15. In a famous series of papers,,, slepian, landau and pollack have given a complete and detailed study of the subject of prolate spheroidal wave functions. Lindquist1, cunhui zhang2, gary glover3, lawrence shepp2, qing. Recurrence relations for prolate spheroidal wave functions. Methods for calculating the wavelet function, clustering the field of wavelet coefficients, and constructing their corresponding optimal rectangular windows in an image are described. This chapter is based in part on abramowitz and stegun 1964, chapter 21 by a.
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. On generalized prolate spheroidal functions preliminary. Prolate spheroidal wave functions pswfs provide a natural and effective tool for computing with bandlimited functions defined on an interval. Spheroidal wave functions become elementary if gammanpi2. Volkmer department of mathematical sciences, university of wisconsinmilwaukee, milwaukee, wisconsin. In this paper we examine the merits of using prolate spheroidal wave functions as basis functions in pseudospectral solution of hyperbolic partial differential. Pdf prolate spheroidal wave functions associated with the.
Prolate spheroidal wave functions department of computer science. The magic of the prolate spheroidal functions in various. Soleski,a new friendly method of computing prolate spheroidal wave func tions and wavelets, appl. Recent investigations of electromagnetic problems in regions bounded by spheroids have led to an mcreased interest in the spheroidal wave functions. The wave equation in prolate spheroidal coordinates is 1 where 2. Fourier expansion of generalized prolate spheroidal wave. Pdf prolate spheroidal wave functions, an introduction. 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 computer science yale. The magic of the prolate spheroidal functions in various setups.
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. Prolate spheroidal scatterer for spherical tem waves. Waveletodd prolate spheroidal wave functions in the. Prolate spheroidal wave functions on a discintegration. Prolate spheroidal wave function as exact solution of the. Pswf prolate spheroidal wave function acronymfinder.
Prolate spheroidal wave functions pswfs are the eigenfunctions of the bandlimited operator in one dimension. Determinant of a matrix involving the prolate spheroidal. They are called oblate spheroidal wave functions if oblate spheroidal coordinates are used and prolate spheroidal wave functions if prolate. We let h b c denotes the space of hankel bandlimited functions given by h b c f.
The 1d case, in which d is an interval, is investigated in 3,5,6. Pdf sampling with prolate spheroidal wave functions. Uniform asymptotic approximations are obtained for the prolate spheroidal functions ps n mx. Let denote the time truncation operator, such that iff x is timelimited within.
The helmholtz differential equation is separable in prolate spheroidal coordinates see also helmholtz differential equationprolate spheroidal coordinates, latitude, longitude, oblate spheroidal coordinates, spherical coordinates. R okhlin, highfrequency asymptotic expansions for certain prolate spheroidal wave functions, j. Abstract generalized prolate spheroidal functions gpsf are the eigen functionsofthetruncatedfouriertransform, restrictedtoddimensional balls in the spatial domain and frequency domain. A generalization of the two dimensional prolate spheroidal. Eigenvalues of spheroidal wave functions are calculated as functions of complex propagation constants. Baum university of new mexico department of electrical and computer engineering albuquerque new mexico 871 abstract this paper considers the scattering of an inhomogeneous spherical tem wave emitted from one of two foci of a prolate spheroid special case of an ellipsoid of revolution. The presentation in 3,5,6 does not handle the numerical aspects of these functions. The twodimensional 2d prolate spheroidal wave function 2d pswf method was previously introduced as an efficient method for trading off between spatial and temporal resolution in magnetic. The orthogonal properties of linear prolate spheroidal wave functions pswfs are exploited to form an orthogonal basis set needed for synthesis. The perturbation theory has been applied to prolate spheroidal models to obtain internal em absorbed power distributions, but the convergence is generally slow and the equations are valid only when the semiaxial lengths of the spheroid are much.
These charts show that the eigenvalues have many branch points in the complex planes of their arguments. On the n dimensional extension of the discrete prolate spheroidal windo w ieee signal processing letters. Osa generalized prolate spheroidal wave functions for. Abstract a simple, efficient and reliable method to extrapolate bandlimited signals varying from lower to higher frequencies is proposed.
On the asymptotic expansion of the spheroidal wave. Robust signal reconstruction using the prolate spherical wave. Approximate formulae for certain prolate spheroidal wave functions. Mathematical tools for bandlimited approximation applied mathematical sciences. These are of great utility in disciplines such as geodesy 28 or cosmology. Prolate spheroidal wave functions, fourier analysis and. They later extended the reconstruction method combining with the tikhonov.
As such, they play an important role in signal processing, fourier analysis, and approximation theory. 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. Prolate spheroidal wave functions pswfs give a very convenient basis for representing and manipulating bandlimited functions restricted to a compact domain d. Assuming nonuniform sampling our procedure can be connected with compressive sensing without complex reconstruction methods. Slepian was also the first to note the connection between pswfs. Spatial smoothing in fmri using prolate spheroidal wave functions martin a. Rmn is the prolate spheroidal radial function, which becomes a spherical bessel function. Spatial smoothing in fmri using prolate spheroidal wave. Combining these two observations and carrying out the elementary but. On the evaluation of prolate spheroidal wave functions and. On the use of discrete prolate spheroidal windows for. First all i would like to tell that this problem arises in gabor theory if one wants to prove that a gabor frame can be generated by multiple prolate spheroidal wave functions i. Approximate formulae for certain prolate spheroidal wave functions valid for large values of both order and bandlimit.
Although these functions cannot be given in closed. Prolate spheroidal wave functions pswfs are the natural basis for representing ban dlimited functions on the interval. 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 and. 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. Prolate spheroidal wave functions pswfs provide a natural and effective tool for. They also find considerable application in other areas such as signal processing and.
Flammer, spheroidal wave functions, 1957 are included as a special case of meixners more general functions. Circular prolate spheroidal wave functions in this section, we prove some desirable properties of the cpswfs. On the asymptotic expansion of the spheroidal wave function. Determinant of a matrix involving the prolate spheroidal wave. Prolate spheroidal wave functions, an introduction to the slepian series and its properties. Prolate spheroid definition of prolate spheroid by the.
The prolate spheroidal wave functions the prolate spheroidal wave functions, pswfs. Spectral methods based on prolate spheroidal wave functions for. Starting from a nonclassical version of the uncertainty principle, they have shown that among the set of bandlimited functions with bandwidth c 0, the most concentrated functions on. Similarly, let denote an ideal lowpass filtering operator, such that iff x is bandlimited within. 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. Slepianprolate spheroidal wave functions, fourier analysis and uncertaintyiv. The energy spectrum is given by the prolate eigenvalues. 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. I have worked for some time on the problem and made some progress. Spectral analysis of the finite hankel transform and. Mathematical tools for bandlimited approximation applied mathematical sciences book 187 kindle edition by osipov, andrei, rokhlin, vladimir, xiao, hong. Two dimensional prolate spheroidal wave functions for mri.
The solution of the schrodinger equation is fundamental for capturing the. The generalized prolate spheroidal wave functions has been recently defined by gupta 4 as the solution of the differential equation 2. Spheroidal wave functions flammer pdf spheroidal wave functions of flammer c. Prolate spheroidal wave functions pswfs play an important role in various areas, from physics e. Pdf prolate spheroidal wave functions associated with. Despite their useful properties in many applications, gpsfs are often replaced by crude approximations. As a result, the energy spectrum of the harmonic oscillator is recovered from the current quantum model as the angular frequency that is to say, the bandlimit tends to infinity. Prolate spheroidal wave functions, an introduction to the. In mathematics, the prolate spheroidal wave functions are a set of functions derived by timelimiting and lowpassing, and a second timelimit operation. These functions 8 mnc, 1, rmnc, result from the solution of the scalar wave equation by separation.
Recently, pswfs have been becoming increasingly popular in various areas in which such functions occur this includes physics e. Generalized prolate spheroidal wave functions springerlink. Prolate spheroidal wave function how is prolate spheroidal. 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. Jul 23, 2016 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. On the n dimensional extension of the discrete prolate. Prolate spheroidal scatterer for spherical tem waves carl e. 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. Spectral analysis of the finite hankel transform and circular. 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 1 t c is a selfadjoint compact operator.
Download it once and read it on your kindle device, pc, phones or tablets. 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. 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 functions on a discintegration and. These finite transforms are more flexible than the fift.
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