Journal ArticleDOI
Prolate Spheroidal Wave Functions, Quadrature, and Interpolation
H Xiao,V Rokhlin,N Yarvin +2 more
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TLDR
In this paper, the authors 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 wavefunctions to construct quadratures, interpolation and differentiation formulae, etc.Abstract:
Polynomials are one of the 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 wavefunctions 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.read more
Citations
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Journal ArticleDOI
Regridding reconstruction algorithm for real-time tomographic imaging
TL;DR: A fast algorithm for tomographic reconstruction based on the Fourier method provides an up to 20-fold performance increase compared with filtered back-projection routines with negligible accuracy degradation.
Journal ArticleDOI
Mathematical concepts of optical superresolution
TL;DR: An overview of some mathematical concepts relevant to superresolution in linear optical systems and properties of bandlimited functions related to both instrumental and computational aspects of superresolution are presented.
Book
A fast algorithm for the calculation of the roots of special functions
TL;DR: A procedure for the determination of the roots of functions satisfying second-order ordinary differential equations, including the classical special functions, is described, based on a combination of the Prufer transform with the classical Taylor series method.
Journal ArticleDOI
A direct solver with O(N) complexity for integral equations on one-dimensional domains
TL;DR: An algorithm for the direct inversion of the linear systems arising from Nyström discretization of integral equations on one-dimensional domains is described, and is closely related to previous work on Hierarchically Semi-Separable matrices.
Journal ArticleDOI
Prolate spheroidal wave functions, an introduction to the Slepian series and its properties
TL;DR: The Slepian series as discussed by the authors is an orthogonal expansion of the spheroidal wave function (PSWF) that is potentially optimal for discontinuous functions such as the square wave among others.
References
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Journal ArticleDOI
Methods of Theoretical Physics. By P.M. Morse and H. Feschbach. 2vols., Pp.xxii, 1978. 120s. each vol. 1953.(McGraw-Hill)
Book
Introduction to Numerical Analysis
Josef Stoer,Roland Bulirsch +1 more
TL;DR: This well written book is enlarged by the following topics: B-splines and their computation, elimination methods for large sparse systems of linear equations, Lanczos algorithm for eigenvalue problems, implicit shift techniques for theLR and QR algorithm, implicit differential equations, differential algebraic systems, new methods for stiff differential equations and preconditioning techniques.
Journal ArticleDOI
Prolate spheroidal wave functions, fourier analysis and uncertainty — II
David Slepian,H. O. Pollak +1 more
TL;DR: In this paper, the authors apply the theory developed in the preceding paper to a number of questions about timelimited and bandlimited signals, and find the signals which do the best job of simultaneous time and frequency concentration.
Journal ArticleDOI
Prolate spheroidal wave functions, fourier analysis, and uncertainty — V: the discrete case
TL;DR: In this article, the authors investigated the extent to which a time series can be concentrated on a finite index set and also have its spectrum concentrated on subinterval of the fundamental period of the spectrum.
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Prolate spheroidal wave functions, fourier analysis and uncertainty — III: The dimension of the space of essentially time- and band-limited signals
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