Approximate formulae for certain prolate spheroidal wave functions valid for large values of both order and band-limit

Rokhlin, Vladimir; Xiao, Han

doi:10.1016/j.acha.2006.05.004

Cited by 54 publications

(76 citation statements)

References 10 publications

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“…∼ n, and therefore, by Theorem 11 in [22], |∂ x ψ c n (0)| ∼ n. Consequently, |β n 1 | shares the same asymptotic behavior with |β n 0 | as in (2.23). Notice that these estimates are more precise than those in Lemma A.2 of [10], which played an important role in the derivations of the main approximation result stated in Theorem 3.2.…”

Section: Lemma 22 For Anymentioning

confidence: 60%

“…(iv) ψ c n (x) has exactly n real distinct roots in the interval (−1, 1). Next, we have from Theorem 12 of Rokhlin and Xiao [22] that for any c > 0,…”

Section: Prolate Spheroidal Wave Functionsmentioning

confidence: 99%

“…Most notably, a series of papers by Slepian et al [25,19,26] and the recent works by Xiao and Rokhlin et al [33,32,24,22] have shown that the PSWFs are a natural and optimal apparatus for approximating bandlimited functions. Recently, there has been a growing interest in developing numerical methods using PSWFs as basis functions, which include in particular the PSWF wavelets [31,30,29] and the PSWF spectral/spectral-element methods [7,8,3,10,18,17,27].…”

Section: Introductionmentioning

confidence: 99%

“…Although there is an increasing number of papers devoted to analytic properties and numerical evaluation of the PSWFs (see, e.g., [33,5,6,21,22,13,34] and the references therein), the approximability of the PSWFs is studied in a limited number of papers. Some error analysis for approximation of bandlimited functions by PSWFs was carried out in the note [24] (also see Xiao's thesis [32]).…”

Section: Introductionmentioning

confidence: 99%

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Analysis of spectral approximations using prolate spheroidal wave functions

Wang

2009

Math. Comp.

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Abstract. In this paper, the approximation properties of the prolate spheroidal wave functions of order zero (PSWFs) are studied, and a set of optimal error estimates are derived for the PSWF approximation of non-periodic functions in Sobolev spaces. These results serve as an indispensable tool for the analysis of PSWF spectral methods. A PSWF spectral-Galerkin method is proposed and analyzed for elliptic-type equations. Illustrative numerical results consistent with the theoretical analysis are also presented.

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Section: Lemma 22 For Anymentioning

confidence: 60%

“…(iv) ψ c n (x) has exactly n real distinct roots in the interval (−1, 1). Next, we have from Theorem 12 of Rokhlin and Xiao [22] that for any c > 0,…”

Section: Prolate Spheroidal Wave Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Analysis of spectral approximations using prolate spheroidal wave functions

Wang

2009

Math. Comp.

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“…The inverse Fourier transform of a series of even Legendre polynomials corresponds to a series of even spherical Bessel functions, that can themselves be expanded in the interval [−1, 1] in terms of even Legendre polynomials [40], [41]. According to this argument, ψ (a)…”

Section: B Frp With Prolate Spheroidal Wavefunctionsmentioning

confidence: 99%

A Forward Regridding Method With Minimal Oversampling for Accurate and Efficient Iterative Tomographic Algorithms

Arcadu

Nilchian

Studer

et al. 2016

IEEE Trans. on Image Process.

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Abstract-Reconstruction of underconstrained tomographic data sets remains a major challenge. Standard analytical techniques frequently lead to unsatisfactory results due to insufficient information. Several iterative algorithms, which can easily integrate a priori knowledge, have been developed to tackle this problem during the last few decades. Most of these iterative algorithms are based on an implementation of the Radon transform that acts as forward projector. This operator and its adjoint, the backprojector, are typically called few times per iteration and represent the computational bottleneck of the reconstruction process. Here, we present a Fourier-based forward projector, founded on the regridding method with minimal oversampling. We show that this implementation of the Radon transform significantly outperforms in efficiency other state-of-the-art operators with O(N 2 log 2 N) complexity. Despite its reduced computational cost, this regridding method provides comparable accuracy to more sophisticated projectors and can, therefore, be exploited in iterative algorithms to substantially decrease the time required for the reconstruction of underconstrained tomographic data sets without loss in the quality of the results.Index Terms-Tomography, iterative reconstruction algorithms, Radon transform, forward and backprojector, regridding method, minimal oversampling, kaiser-bessel, prolate-spheroidal wavefunctions.

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Generalized prolate spheroidal wave functions for offset linear canonical transform in Clifford analysis

Kou

Morais

Zhang

2012

Math Methods in App Sciences

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Prolate spheroidal wave functions (PSWFs) possess many remarkable properties. They are orthogonal basis of both square integrable space of finite interval and the Paley–Wiener space of bandlimited functions on the real line. No other system of classical orthogonal functions is known to obey this unique property. This raises the question of whether they possess these properties in Clifford analysis. The aim of the article is to answer this question and extend the results to more flexible integral transforms, such as offset linear canonical transform. We also illustrate how to use the generalized Clifford PSWFs (for offset Clifford linear canonical transform) we derive to analyze the energy preservation problems. Clifford PSWFs is new in literature and has some consequences that are now under investigation. Copyright © 2012 John Wiley & Sons, Ltd.

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Approximate formulae for certain prolate spheroidal wave functions valid for large values of both order and band-limit

Cited by 54 publications

References 10 publications

Analysis of spectral approximations using prolate spheroidal wave functions

Analysis of spectral approximations using prolate spheroidal wave functions

A Forward Regridding Method With Minimal Oversampling for Accurate and Efficient Iterative Tomographic Algorithms

Generalized prolate spheroidal wave functions for offset linear canonical transform in Clifford analysis

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