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Mathematics > Numerical Analysis

arXiv:2206.00448 (math)
[Submitted on 1 Jun 2022]

Title:A fast algorithm for the inversion of Abel's transform

Authors:Enrico De Micheli
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Abstract: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.
Comments: 19 pages, 4 figures
Subjects: Numerical Analysis (math.NA); Classical Analysis and ODEs (math.CA)
MSC classes: 45E10 (Primary) 44A15, 65R32 (Secondary)
Cite as: arXiv:2206.00448 [math.NA]
  (or arXiv:2206.00448v1 [math.NA] for this version)
  https://doi.org/10.48550/arXiv.2206.00448
Journal reference: Applied Mathematics and Computation, 301 (2017), 12-24
Related DOI: https://doi.org/10.1016/j.amc.2016.12.009

Submission history

From: Enrico De Micheli [view email]
[v1] Wed, 1 Jun 2022 12:30:22 UTC (181 KB)
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