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The application of orthonormal basis functions such as Prolate Spheroidal Wave Functions (PSWF) for accurate source modeling in radio astronomy has been comprehensively studied. They are of great importance for high fidelity, high dynamic range imaging with new radio telescopes as well as conventional ones. But the construction of PSWF is computationally expensive compared to other closed form basis functions. In this paper, we suggest a solution to reduce its computational cost by more efficient construction of the matrix kernel which relates the image domain to visibility (or Fourier) domain. Radio astronomical images are mostly represented using a regular grid of rectangular pixels. This is required for efficient storage and display purposes and moreover, comes naturally as a by product of the Fast Fourier Transform (FFT) in imaging. We propose the use of Delaunay triangulation as opposed to regular gridding of an image for a finer selection of the region of interest (signal support) during the PSWF kernel construction. We show that the computational efficiency improves without loss of information. Once the PSWF basis is constructed using the irregular grid, we revert back to the regular grid by interpolation and thereafter, conventional imaging techniques can be applied.

We introduce a new class of numerical differentiation schemes constructed for the efficient solution of time-dependent PDEs that arise in wave phenomena. The schemes are constructed via the prolate spheroidal wave functions (PSWFs). Compared to existing differentiation schemes based on orthogonal polynomials, the new class of differentiation schemes requires fewer points per wavelength to achieve the same accuracy when it is used to approximate derivatives of bandlimited functions. In addition, the resulting differentiation matrices have spectral radii that grow asymptotically as m for the case of first derivatives, and m sq for second derivatives, with m being the dimensions of the matrices. The above results mean that the new class of differentiation schemes is more efficient in the solution of time-dependent PDEs compared to existing schemes such as the Chebyshev collocation method. The improvements are particularly prominent in large-scale time-dependent PDEs, in which the solutions contain large numbers of wavelengths in the computational domains.

For fixed c, Prolate Spheroidal Wave Functions $\psi_{n, c}$ form a basis with remarkable properties for the space of band-limited functions with bandwith $c$. They have been largely studied and used after the seminal work of Slepian. Recently, they have been used for the approximation of functions of the Sobolev space $H^s([-1,1])$. The choice of $c$ is then a central issue, which we address. Such functions may be seen as the restriction to $[-1,1]$ of almost time-limited and band-limited functions, for which PSWFs expansions are still well adapted. To be able to give bounds for the speed of convergence one needs uniform estimates in $n$ and $c$. To progress in this direction, we push forward the WKB method and find uniform approximation of $\psi_{n, c}$ in terms of the Bessel function $J_0$ while only point-wise asymptotic approximation was known up to now. Many uniform estimates can be deduced from this analysis. Finally, we provide the reader with numerical examples that illustrate in particular the problem of the choice of c.

For fixed c, Prolate Spheroidal Wave Functions $\psi_{n, c}$ form a basis with remarkable properties for the space of band-limited functions with bandwith $c$. They have been largely studied and used after the seminal work of Slepian. Recently, they have been used for the approximation of functions of the Sobolev space $H^s([-1,1])$. The choice of $c$ is then a central issue, which we address. Such functions may be seen as the restriction to $[-1,1]$ of almost time-limited and band-limited functions, for which PSWFs expansions are still well adapted. To be able to give bounds for the speed of convergence one needs uniform estimates in $n$ and $c$. To progress in this direction, we push forward the WKB method and find uniform approximation of $\psi_{n, c}$ in terms of the Bessel function $J_0$ while only point-wise asymptotic approximation was known up to now. Many uniform estimates can be deduced from this analysis. Finally, we provide the reader with numerical examples that illustrate in particular the problem of the choice of c.

We prove a weak version of Hardy's uncertainty principle using properties of the prolate spheroidal wave functions (PSWFs). We describe the eigenvalues of the sum of a time limiting operator and a band limiting operator acting on L2(R). A weak version of Hardy's uncertainty principle follows from the asymptotic behavior of the largest eigenvalue as the time limit and the band limit approach infinity. An asymptotic formula for this eigenvalue is obtained from its well-known counterpart for the prolate integral operator.

Bell System Technical Journal, 57: 5. May-June 1978 pp 1371-1430. Prolate Spheroidal Wave Functions, Fourier Analysis, and Uncertainty - V: The Discrete Case. (Slepian, D.)

Bell System Technical Journal, 57: 5. May-June 1978 pp 1371-1430. Prolate Spheroidal Wave Functions, Fourier Analysis, and Uncertainty - V: The Discrete Case. (Slepian, D.)

In order to produce high dynamic range images in radio interferometry, bright extended sources need to be removed with minimal error. However, this is not a trivial task because the Fourier plane is sampled only at a finite number of points. The ensuing deconvolution problem has been solved in many ways, mainly by algorithms based on CLEAN. However, such algorithms that use image pixels as basis functions have inherent limitations and by using an orthonormal basis that span the whole image, we can overcome them. The construction of such an orthonormal basis involves fine tuning of many free parameters that define the basis functions. The optimal basis for a given problem (or a given extended source) is not guaranteed. In this paper, we discuss the use of generalized prolate spheroidal wave functions as a basis. Given the geometry (or the region of interest) of an extended source and the sampling points on the visibility plane, we can construct the optimal basis to model the source. Not only does this gives us the minimum number of basis functions required but also the artifacts outside the region of interest are minimized.

The spin-weighted spheroidal equations in the case s=1/2 is thoroughly studied in the paper by means of the perturbation method in supersymmetry quantum mechanics. The first-five terms of the super-potential in the series of the parameter beta are given. The general form of the nth term of the superpotential is also obtained, which could derived from the previous terms W_{k}, k

The perturbation method in supersymmetric quantum mechanics (SUSYQM) is used to study the spheroidal wave functions' eigenvalue problem. Expanding the super-potential in series of the parameter alpha, the first order term of ground eigen-value and the eigen-function are gotten. In the paper, the very excellent results are that all the first two terms approximation on eigenfunctions obtained are in closed form. They give useful information for the involved physical problems in application of spheroidal wave functions.

The perturbation method in supersymmetric quantum mechanics (SUSYQM) is used to study the spheroidal wave functions' recurrence relations, which are revealed by the shape-invariance property of the super-potential. The super-potential is expanded by the parameter alpha and could be gotten by approximation method. Up to the first order, it has the shape-invariance property and the excited spheroidal wave functions are gotten. Also, all the first term eigenfunctions obtained are in closed form. They are advantageous to investigating for involved physical problems of spheroidal wave function.

The prolate spheroidal wave functions, which are a special case of the spheroidal wave functions, possess a very surprising and unique property [6]. They are an orthogonal basis of both $L^2(-1,1)$ and the Paley-Wiener space of bandlimited functions. They also satisfy a discrete orthogonality relation. No other system of classical orthogonal functions is known to possess this strange property. We prove that there are new systems possessing this property in $q$-Fourier analysis. As application we give a new sampling formula with $q^n$ as sampling points, where 0 < q < 1.

The spheroidal wave functions, which are the solutions to the Helmholtz equation in spheroidal coordinates, are notoriously difficult to compute. Because of this, practically no programming language comes equipped with the means to compute them. This makes problems that require their use hard to tackle. We have developed computational software for calculating these special functions. Our software is called spheroidal and includes several novel features, such as: using arbitrary precision arithmetic; adaptively choosing the number of expansion coefficients to compute and use; and using the Wronskian to choose from several different methods for computing the spheroidal radial functions to improve their accuracy. There are two types of spheroidal wave functions: the prolate kind when prolate spheroidal coordinates are used; and the oblate kind when oblate spheroidal coordinate are used. In this paper, we describe both, methods for computing them, and our software. We have made our software freely available on our webpage.

Bell System Technical Journal, 41: 4. July 1962 pp 1295-1336. Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty -- III: The Dimension of the Space of Essentially Time- and Band-Limited Signals. (Landau, H.J.; Pollak, H.O.)

Bell System Technical Journal, 40: 1. January 1961 pp 43-63. Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - I. (Slepian, D.; Pollak, H.O. 65-84)

Bell System Technical Journal, 43: 6. November 1964 pp 3009-3057. Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - IV: Extensions to Many Dimensions; Generalized Prolate Spheroidal Functions. (Slepian, David)

A new set of vector solutions to Maxwell's equations based on solutions to the wave equation in spheroidal coordinates allows laser beams to be described beyond the paraxial approximation. Using these solutions allows us to calculate the complete first-order corrections in the short-wavelength limit to eigenmodes and eigenfrequencies in a Fabry-Perot resonator with perfectly conducting mirrors. Experimentally relevant effects are predicted. Modes which are degenerate according to the paraxial approximation are split according to their total angular momentum. This includes a splitting due to coupling between orbital angular momentum and spin angular momentum.

In this paper, we introduce a new set of functions, which have the property of the completeness over a finite and infinite intervals. This family of functions, denoted for simplicity GOSWFs, are a generalization of the oblate spheroidal wave functions. They generalize also the Jacobi polynomials in some sens. The GOSWFs are nothing but the eigenfunctions of the finite weighted bilateral Laplace transform $\mathcal{F}_c^{(\alpha,\beta)}.$ We compute this functions by two methods: In the first one we use a differential operator $\mathcal{D}$ which commutes with $\mathcal{F}_c^{(\alpha,\beta)}.$ In the second one we use the Gaussian quadrature method. As an application, we use the GOSWFs to invert the finite bilateral Laplace transform. We also use the GOSWFs to approximate bilateral weighted Laplace bandlimited functions and we show that they are more advantageous then other classical basis of $L^2((-1,1),(1-x)^\alpha(1+x)^\beta)dx$. Finally, we provide the reader by some numerical examples that illustrate the theoretical results.

Bell System Technical Journal, 44: 8. October 1965 pp 1745-1759. Eigenvalues Associated with Prolate Spheroidal Wave Functions of Zero Order. (Slepian, David; Sonnenblick, Estelle)

As demonstrated by Slepian et. al. in a sequence of classical papers, prolate spheroidal wave functions (PSWFs) provide a natural and efficient tool for computing with bandlimited functions defined on an interval. Recently, PSWFs have been becoming increasingly popular in various areas in which such functions occur - this includes physics (e.g. wave phenomena, fluid dynamics), engineering (signal processing, filter design), etc. To use PSWFs as a computational tool, one needs fast and accurate numerical algorithms for the evaluation of PSWFs and related quantities, as well as for the construction of corresponding quadrature rules, interpolation formulas, etc. During the last 15 years, substantial progress has been made in the design of such algorithms. However, many of the existing algorithms tend to be relatively slow when $c$ is large (e.g. c>10^4). In this paper, we describe several numerical algorithms for the evaluation of PSWFs and related quantities, and design a class of PSWF-based quadratures for the integration of bandlimited functions. While the analysis is somewhat involved and will be published separately, the resulting numerical algorithms are quite simple and efficient in practice. For example, the evaluation of the $n$th eigenvalue of the prolate integral operator requires $O(n+c \cdot \log c)$ operations; the construction of accurate quadrature rules for the integration (and associated interpolation) of bandlimited functions with band limit $c$ requires $O(c)$ operations. All algorithms described in this paper produce results essentially to machine precision. Our results are illustrated via several numerical experiments.

Polynomials are one of principal tools of classical numerical analysis. When a function needs to be interpolated, integrated, differentiated, etc., it is assumed to be approximated by a polynomial of a certain fixed order (though the polynomial is almost never constructed explicitly), and a treatment appropriate to such a polynomial is applied. We introduce analogous techniques based on the assumption that the function to be dealt with is band-limited, and use the well-developed apparatus of Prolate Spheroidal Wave Functions to construct quadratures, interpolation and differentiation formulae, etc. for band-limited functions. Since band-limited functions are often encountered in physics, engineering, statistics, etc. the apparatus we introduce appears to be natural in many environments. Our results are illustrated with several numerical examples.

Polynomials are one of principal tools of classical numerical analysis. When a function needs to be interpolated, integrated, differentiated, etc., it is assumed to be approximated by a polynomial of a certain fixed order (though the polynomial is almost never constructed explicitly), and a treatment appropriate to such a polynomial is applied. We introduce analogous techniques based on the assumption that the function to be dealt with is band-limited, and use the well-developed apparatus of Prolate Spheroidal Wave Functions to construct quadratures, interpolation and differentiation formulae, etc. for band-limited functions. Since band-limited functions are often encountered in physics, engineering, statistics, etc. the apparatus we introduce appears to be natural in many environments. Our results are illustrated with several numerical examples.

Bell System Technical Journal, 40: 1. January 1961 pp 65-84. Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - II. (Landau, H.J.; Pollak, H.O.)

One of the fundamental problems in communications is finding the energy distribution of signals in time and frequency domains. It should, therefore, be of great interest to find the most energy concentration hypercomplex signal. The present paper finds a new kind of hypercomplex signals whose energy concentration is maximal in both time and frequency under quaternionic Fourier transform. The new signals are a generalization of the prolate spheroidal wave functions (also known as Slepian functions) to quaternionic space, which are called quaternionic prolate spheroidal wave functions. The purpose of this paper is to present the definition and properties of the quaternionic prolate spheroidal wave functions and to show that they can reach the extreme case in energy concentration problem both from the theoretical and experimental description. In particular, these functions are shown as an effective method for bandlimited signals extrapolation problem.

The perturbation method in supersymmetric quantum mechanics (SUSYQM) is used to study whether the spheroidal equations have the shape-invariance property. Expanding the super-potential term by term in the parameter alpha and solving it, we find that the superpotential loses its shape-invariance property upon to the second term. This first means that we could not solve the spheroidal problems by the SUSQM; further it is not unreasonable to say they are non-solvable in some sense.

The perturbation method in supersymmetric quantum mechanics (SUSYQM) is used to study whether the spheroidal equations have the shape-invariance property. Expanding the super-potential term by term in the parameter alpha and solving it, we find that the superpotential loses its shape-invariance property upon to the second term. This first means that we could not solve the spheroidal problems by the SUSQM; further it is not unreasonable to say they are non-solvable in some sense.

The perturbation method in supersymmetric quantum mechanics (SUSYQM) is used to study whether the spheroidal equations have the shape-invariance property. Expanding the super-potential term by term in the parameter alpha and solving it, we find that the superpotential loses its shape-invariance property upon to the second term. This first means that we could not solve the spheroidal problems by the SUSQM; further it is not unreasonable to say they are non-solvable in some sense.