Algorithms to numerically evaluate the Hankel transform

Cree, Michael J.; Bones, Philip J.

doi:10.1016/0898-1221(93)90081-6

Cited by 34 publications

(26 citation statements)

References 30 publications

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“…Here we use transforms which are available in Matlab. 3 The correlation is adjusted so that the last value is zero, as expected for the Hankel transform [36]. The results from two other simulations are shown in Figures (11-12).…”

Section: Preprints (Wwwpreprintsorg) | Not Peer-reviewed | Postedmentioning

confidence: 94%

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Calculating Energy Spectra from Drifters

LaCasce

2016

Preprint

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Abstract:The relations between the kinetic energy spectrum and the second order longitudinal structure function in two dimensions are derived, and several examples are considered. The forward conversion (from spectrum to structure function) is illustrated first with idealized power law spectra, representing turbulent inertial ranges. The forward conversion is also applied to the kinetic energy spectrum of Nastrom and Gage (1985), derived from aircraft data in the upper troposphere, and the result agrees well with the longitudinal structure function of Lindborg (1999), using the same data. The inverse conversion (from structure function to spectrum) is tested with data from 2D turbulence simulations. When applied to the theoretical structure function (derived from the forward conversion of the spectrum), the result closely resembles the original spectrum, except at the largest wavenumbers. However the inverse conversion is much less successful when applied to the structure function obtained from pairs of particles in the flow. This is because the inverse conversion favors large pair separations, which are typically noisy with particle data. Fitting the structure function to a polynomial improves the result, but not sufficiently to distinguish the correct inertial range dependencies. Furthermore the inversion of non-local spectra is largely unsuccessful. Thus it appears that focusing on structure functions with Lagrangian data is preferable to estimating spectra.

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Section: Preprints (Wwwpreprintsorg) | Not Peer-reviewed | Postedmentioning

confidence: 94%

“…2 [36] discuss the numerical evaluation of Hankel transforms. Here we use transforms which are available in Matlab.…”

Section: Preprints (Wwwpreprintsorg) | Not Peer-reviewed | Postedmentioning

confidence: 99%

Calculating Energy Spectra from Drifters

LaCasce

2016

Preprint

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“…If the strip of convergence of the Mellin or two-sided Laplace transform includes the imaginary axis , then the Mellin and inverse Mellin transforms can be replaced by a Fourier and an inverse Fourier transform, providing the basis for FFT-based algorithms [7]- [10]. However, the need to have sampled on an exponential grid is a severe disadvantage since it amounts to a coarse undersampling of the tail away from the origin of the function [6].…”

Section: Direct Mellin Approachmentioning

confidence: 99%

“…Over the past 25 years, a number of algorithms for the numerical evaluation of the Hankel transform have been reported in the literature. For an overview of these algorithms and their numerical complexity, see [6]. Except for the obvious but inefficient numerical quadrature method, all these algorithms can be cast into three general classes.…”

Section: Introductionmentioning

confidence: 99%

Fast Hankel transform by fast sine and cosine transforms: the Mellin connection

Knockaert

2000

IEEE Trans. Signal Process.

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Abstract-TheHankel transform of a function by means of a direct Mellin approach requires sampling on an exponential grid, which has the disadvantage of coarsely undersampling the tail of the function. A novel modified Hankel transform procedure that does not require exponential sampling is presented. The algorithm proceeds via a three-step Mellin approach to yield a decomposition of the Hankel transform into a sine, a cosine, and an inversion transform, which can be implemented by means of fast sine and cosine transforms.Index Terms-Hankel transform, Mellin transform, sine and cosine transform.

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“…, ρ n are usually chosen to be the first n positive roots of J ν . Previous methods for numerically computing Fourier-Bessel transforms include using an exponential change of variables, asymptotic expansions of Bessel functions, and projection techniques (see [5], [12], and [17]). These algorithms often had limitations on the choices for ρ in equation (1.13) (due to the requirement of equispaced points for FFTs), the requirement of sampling g on an exponential grid, and on the order ν of the transform.…”

Section: Introductionmentioning

confidence: 99%

A New Class of Analysis-Based Fast Transforms

Rokhlin¹,

O’Neil²

2007

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We introduce a new approach to the rapid numerical application to arbitrary vectors of certain types of linear operators. Inter alia, our scheme is applicable to many classical integral transforms, and to the expansions associated with most families of classical special functions; among the latter are Bessel functions, Legendre, Hermite, and Laguerre polynomials, Spherical Harmonics, Prolate Spheroidal Wave functions, and a number of others. In all these cases, the CPU time requirements of our algorithm are of the order O(n log n), where n is the size of the matrix to be applied. The performance of our algorithm is illustrated via a number of numerical examples.A new class of analysis-based fast transforms

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Algorithms to numerically evaluate the Hankel transform

Cited by 34 publications

References 30 publications

Calculating Energy Spectra from Drifters

Calculating Energy Spectra from Drifters

Fast Hankel transform by fast sine and cosine transforms: the Mellin connection

A New Class of Analysis-Based Fast Transforms

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