Prolate spheroidal wave functions pswfs play an important role in various areas, from physics e. In this paper, we give new characterizations for the eigenvalues of the prolate wave equation as limits of the zeros of some families of. Bandlimited signal extrapolation using prolate spheroidal. Prolate spheroidal wave functions, fourier analysis and.
Generalized prolate spheroidal wave functions for offset. These functions, also known as slepian prolate functions, were bandlimited and exhibited interesting orthogonality relations. Uniform asymptotic approximations are obtained for the prolate spheroidal functions ps n mx. A technique is presented for the calculation of the oblate and prolate spheroidal wave equation eigenvalues and eigenfunctions. Rmn is the prolate spheroidal radial function, which becomes a spherical bessel function. Now at university of bucharest, departmentof physics, bucharestmagurele, romania. Separation of variables of the wave equation in spheroidal coordinates results in a differential equation that is satis. Formulas are obtained in detail for the prolate spheroidal wave functions. The same method would work for a region on the sphere bounded by two parallels. Spectral methods based on prolate spheroidal wave functions. In mathematics, the spheroidal wave equation is given by. Prolate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two smallest principal axes are equal in length.
On the asymptotic expansion of the spheroidal wave function. Interpolation orthogonal expansion prolate spheroidal wave function. An op1q algorithm for the numerical evaluation of the. It is a generalization of the mathieu differential equation. Prolate spheroidal scatterer for spherical tem waves. The solution of the scalar wave equation strongly depends on the objects shape and exact solutions only exist for a few cases. This therefore causes the difficulty of obtaining rigorous. Discrete prolate spheroidal wave functions and interpolation. 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. Prolate spheroidal wave functions pswf, the onedimensional case of. Dunster san diego state university san diego, ca 92182 usa abstract. Prolate spheroidal wave functions, an introduction to the slepian series and its properties. The eigenvalue problem is cast in matrix form and a tridiagonal. A complete set of bandlimited functions is described which possesses the curious property of being orthogonal over a given finite interval as well as.
We explore a number of issues related to this problem, including the effective computation of the corresponding slepzan functions for a polar cap on the surface of the earth. A computational method to calculate the exact solution for. Prolate spheroidal wave function, sampling,paleywinner. On the asymptotic expansion of the spheroidal wave. In this paper we exploit the orthogonal properties of prolate spheroidal wave functions pswf in the form of a new orthogonal expansion which we have named the slepian series. Schrodinger equation in prolate spheroidal coordinates, and the separation of linearized perturbation.
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. Discrete prolate spheroidal sequences and wave functions. Prolate spheroidal coordinates can be used to solve various partial differential equations in which the boundary conditions match its symmetry and shape, such as solving for a field produced by two centers. The differential equations given above for the oblate radial and angular wave functions can be obtained from the corresponding equations for the prolate spheroidal wave functions by the substitution of for and for.
Examples are saturns satellites mimas, enceladus, and tethys and uranus satellite miranda. Teukolskys equations governing perturbations of the kerr metric are generalized spheroidal wave equations. Some examples are presented in the numerical results. The prolate spheroidal wave equation pswe is transformed, using suitable mappings, into three different canonical forms which resemble the jacobi, laguerre and the hermite differential equations. The prolate radial spheroidal wave functions appear in a wide press, stanford, 1957 russian translation, 1962. Despite their useful properties in many applications, gpsfs are often replaced by crude approximations. Publishers pdf, also known as version of record includes final page, issue and. The prolate and oblate spheroidal coordinates are two systems in which the scalar wave equations are separable but the vector wave functions are not separable.
The polynomials rn,n satisfy the differential equation. One can check that pswfs satisfy the following equation. Stokes phenomenon for the prolate spheroidal wave equation. Abstract generalized prolate spheroidal functions gpsf are the eigen functionsofthetruncatedfouriertransform, restrictedtoddimensional balls in the spatial domain and frequency domain. Since helmholtz equation is separable in prolate oblate spheroidal coordinates, its exact solution for the scattered.
The prolate spheroidal wave functions can be characterized as the eigenfunctions of a differential operator of order 2. In this paper, we introduce the prolate spheroidal wave functions pswfs of real order. New characterizations for the eigenvalues of the prolate spheroidal. They are eigenfunctions of a timelimiting operation followed by a lowpassing operation. In this paper, we give new characterizations for the eigenvalues of the prolate wave equation as limits of the zeros of some families of polynomials. Pdf sampling with prolate spheroidal wave functions.
Prolate spheroidal wave functions, an introduction to the. Recurrence relations for prolate spheroidal wave functions. The prolate spheroidal coordinate system helmholm equation is not. Prolate spheroidal coordinates can be used to solve various partial differential equations in which the boundary conditions match its symmetry and shape, such as solving for a field produced by two centers, which are taken as the foci on the z axis. The magic of the prolate spheroidal functions in various setups. Prolate spheroidal wave functions pswfs provide a natural and effective tool for computing with bandlimited. Software for computing the spheroidal wave functions using arbitrary precision arithmetic ross adelman, nail a.
Software for computing the spheroidal wave functions. A spheroid has circular symmetry if the ellipse is rotated about its major axis, the result is a prolate elongated spheroid, shaped like an american football or rugby ball. The magic of the prolate spheroidal functions in various. Prolate spheroidal wave functions department of computer science.
The prolate spheroid is the approximate shape of the ball in several sports, such as in rugby football. 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. Pdf prolate spheroidal wave functions, an introduction to. Pdf prolate spheroidal wave functions, an introduction. Solutions to a generalized spheroidal wave equation. New characterizations for the eigenvalues of the prolate. Pdf chromatic series with prolate spheroidal wave functions. They are normalized versions of the solutions to helmholtz wave equation 1 in spheroidal coordinates.
The transition matrix for acoustic and elastic wave. For, the angular spheroidal wave functions can be expanded as a series of legendre functions. A spheroid, or ellipsoid of revolution, is a quadric surface obtained by rotating an ellipse about one of its principal axes. Several moons of the solar system approximate prolate spheroids in shape, though they are actually triaxial ellipsoids. In mathematics, the prolate spheroidal wave functions pswf are a set of orthogonal bandlimited functions. Asymptotics of prolate spheroidal wave functions t. Prolate spheroidal wave function as exact solution of the schrodinger. Ball prolate spheroidal wave functions in arbitrary dimensions. We set the azimuthal number m in the spheroidal equation equal to 12 and factor out of the solution the term. This makes them bandlimited functions in a paleywiener space, but because they are. It is a very useful identity for the derivation of the orthogonality 2. Keywords prolate spheroidal wave equation classical orthogonal polynomials.
The prolate spheroidal wave functions pswfs originally appeared during the resolution of the helmholtz equation in the spheroidal coordinate system by variables separation 16. Prolate spheroidal scatterer for spherical tem waves carl e. Software for computing the spheroidal wave functions using. Prolate spheroidal wave functions pswfs arise as solutions of an integral equation. Accurate calculation of the modified mathieu and spheroidal. Then we develop a stap algorithm which computes the clutter subspace using the geometry of the problem rather than data and utilizes the blockdiagonal structure of the jammer covariance matrix. We empirically show that the slepian series is potentially optimal over more conventional orthogonal expansions for discontinuous functions such as the square wave among. Pdf the prolate spheroidal wave functions pswfs are used in. Theory and computation of the spheroidal wave functions. Highly accurate pseudospectral approximations of the. Highly accurate pseudospectral approximations of the prolate. This makes them bandlimited functions in a paleywiener space, but because they are also solutions to a sturm.
739 1105 689 278 1583 725 758 1505 212 1467 438 351 489 1195 871 1040 1506 385 1556 1570 1520 1467 344 537 1196 1559 1294 866 1277 677 1452 690 138 963 1593 1520 711 184 973 347 1257 743 151 1068 770 467 95