The prolate radial spheroidal wave functions appear in a wide press, stanford, 1957 russian translation, 1962. Discrete prolate spheroidal wave functions and interpolation. Discrete prolate spheroidal sequences and wave functions. It is a generalization of the mathieu differential equation. 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. We empirically show that the slepian series is potentially optimal over more conventional orthogonal expansions for discontinuous functions such as the square wave among. Highly accurate pseudospectral approximations of the. Bandlimited signal extrapolation using prolate spheroidal. Separation of variables of the wave equation in spheroidal coordinates results in a differential equation that is satis. 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. A computational method to calculate the exact solution for. Prolate spheroidal wave functions pswfs arise as solutions of an integral equation. In mathematics, the prolate spheroidal wave functions pswf are a set of orthogonal bandlimited functions. The eigenvalue problem is cast in matrix form and a tridiagonal.
Now at university of bucharest, departmentof physics, bucharestmagurele, romania. 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. In this paper, we give new characterizations for the eigenvalues of the prolate wave equation as limits of the zeros of some families of. On the asymptotic expansion of the spheroidal wave. A technique is presented for the calculation of the oblate and prolate spheroidal wave equation eigenvalues and eigenfunctions. The prolate spheroidal wave functions pswfs originally appeared during the resolution of the helmholtz equation in the spheroidal coordinate system by variables separation 16. This makes them bandlimited functions in a paleywiener space, but because they are. For, the angular spheroidal wave functions can be expanded as a series of legendre functions. Stokes phenomenon for the prolate spheroidal wave equation.
Rmn is the prolate spheroidal radial function, which becomes a spherical bessel function. Prolate spheroidal wave functions pswfs play an important role in various areas, from physics e. Interpolation orthogonal expansion prolate spheroidal wave function. 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. Abstract generalized prolate spheroidal functions gpsf are the eigen functionsofthetruncatedfouriertransform, restrictedtoddimensional balls in the spatial domain and frequency domain. Dunster san diego state university san diego, ca 92182 usa abstract. An op1q algorithm for the numerical evaluation of the. Prolate spheroidal wave function, sampling,paleywinner. 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. Formulas are obtained in detail for the prolate spheroidal wave functions. 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. Highly accurate pseudospectral approximations of the prolate. Solutions to a generalized spheroidal wave equation. Accurate calculation of the modified mathieu and spheroidal.
Prolate spheroidal scatterer for spherical tem waves. On the asymptotic expansion of the spheroidal wave function. These functions, also known as slepian prolate functions, were bandlimited and exhibited interesting orthogonality relations. Ball prolate spheroidal wave functions in arbitrary dimensions. Prolate spheroidal wave functions, fourier analysis and. In mathematics, the spheroidal wave equation is given by. This therefore causes the difficulty of obtaining rigorous. Some examples are presented in the numerical results. In this paper, we introduce the prolate spheroidal wave functions pswfs of real order. Generalized prolate spheroidal wave functions for offset. One can check that pswfs satisfy the following equation.
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. 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. Teukolskys equations governing perturbations of the kerr metric are generalized spheroidal wave equations. Pdf the prolate spheroidal wave functions pswfs are used in. The magic of the prolate spheroidal functions in various setups. There are several ways to generate the function set, each having its advantages and disadvantages in. Spectral methods based on prolate spheroidal wave functions.
Schrodinger equation in prolate spheroidal coordinates, and the separation of linearized perturbation. Examples are saturns satellites mimas, enceladus, and tethys and uranus satellite miranda. Prolate spheroidal wave functions department of computer science. 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.
Several moons of the solar system approximate prolate spheroids in shape, though they are actually triaxial ellipsoids. They are eigenfunctions of a timelimiting operation followed by a lowpassing operation. They are normalized versions of the solutions to helmholtz wave equation 1 in spheroidal coordinates. Software for computing the spheroidal wave functions using. Since helmholtz equation is separable in prolate oblate spheroidal coordinates, its exact solution for the scattered.
The transition matrix for acoustic and elastic wave. The prolate spheroidal wave functions can be characterized as the eigenfunctions of a differential operator of order 2. The polynomials rn,n satisfy the differential equation. Prolate spheroidal wave function as exact solution of the. The solution of the scalar wave equation strongly depends on the objects shape and exact solutions only exist for a few cases. Asymptotics of prolate spheroidal wave functions t. We set the azimuthal number m in the spheroidal equation equal to 12 and factor out of the solution the term. It is a very useful identity for the derivation of the orthogonality 2.
Prolate spheroidal wave functions, an introduction to the slepian series and its properties. New characterizations for the eigenvalues of the prolate spheroidal. Keywords prolate spheroidal wave equation classical orthogonal polynomials. Prolate spheroidal wave functions pswf, the onedimensional case of. Pdf sampling with prolate spheroidal wave functions. Prolate spheroidal wave functions pswfs provide a natural and effective tool for computing with bandlimited. 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. 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. Prolate spheroidal wave functions, an introduction to the.
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. This makes them bandlimited functions in a paleywiener space, but because they are also solutions to a sturm. Recurrence relations for prolate spheroidal wave functions. Theory and computation of the spheroidal wave functions. Despite their useful properties in many applications, gpsfs are often replaced by crude approximations. New characterizations for the eigenvalues of the prolate. Software for computing the spheroidal wave functions using arbitrary precision arithmetic ross adelman, nail a. A complete set of bandlimited functions is described which possesses the curious property of being orthogonal over a given finite interval as well as.
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