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Legendre Series Analysis and Computation via Composed Abel-Fourier Transform
Version 1
: Received: 19 May 2023 / Approved: 22 May 2023 / Online: 22 May 2023 (12:50:07 CEST)
A peer-reviewed article of this Preprint also exists.
De Micheli, E. Legendre Series Analysis and Computation via Composed Abel–Fourier Transform. Symmetry 2023, 15, 1282. De Micheli, E. Legendre Series Analysis and Computation via Composed Abel–Fourier Transform. Symmetry 2023, 15, 1282.
Abstract
We prove that the Legendre coefficients associated with a function f(x) can be represented as the Fourier coefficients of a suitable Abel-type transform of the function itself. Thus, the computation of N Legendre coefficients can be performed efficiently in O(NlogN) operations by means of a single Fast Fourier Transform of the Abel-type transform of f(x). We also show how the symmetries associated with the Abel-type transform can be exploited to further reduce the computational complexity. The dual problem of calculating the sum of Legendre expansions is also considered. We prove that a Legendre series can be written as the Abel transform of a suitable Fourier series. This fact allows us to state an efficient algorithm for the evaluation of Legendre expansions. Finally, numerical tests are presented to exemplify and confirm the theoretical results.
Keywords
Legendre coefficients; Fourier coefficients; Legendre expansion; Abel transform
Subject
Computer Science and Mathematics, Analysis
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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