A fast algorithm for the inversion of Abel’s transform

We present a new algorithm for the computation of the inverse Abel transform, a problem which emerges in many areas of physics and engineering. We prove that the Legendre coefficients of a given function coincide with the Fourier coefficients of a suitable periodic function associated with its Abel...

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Bibliographic Details
Published in:Applied mathematics and computation Vol. 301; pp. 12 - 24
Main Author: De Micheli, Enrico
Format: Journal Article
Language:English
Published: Elsevier Inc 15.05.2017
Subjects:
Abel inversion
Inverse problems
Legendre polynomials
Plasma emission coefficients
Radio occultation
Stability estimates
Radio occultation
Plasma emission coefficients
Inverse problems
Abel inversion
Legendre polynomials
Stability estimates
ISSN:0096-3003, 1873-5649
Online Access:Get full text
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Summary:We present a new algorithm for the computation of the inverse Abel transform, a problem which emerges in many areas of physics and engineering. We prove that the Legendre coefficients of a given function coincide with the Fourier coefficients of a suitable periodic function associated with its Abel transform. This allows us to compute the Legendre coefficients of the inverse Abel transform in an easy, fast and accurate way by means of a single Fast Fourier Transform. The algorithm is thus appropriate also for the inversion of Abel integrals given in terms of samples representing noisy measurements. Rigorous stability estimates are proved and the accuracy of the algorithm is illustrated also by some numerical experiments.
ISSN:0096-3003
1873-5649
DOI:10.1016/j.amc.2016.12.009