Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

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

Comments (0)

We encourage comments and feedback from a broad range of readers. See criteria for comments and our Diversity statement.

Leave a public comment
Send a private comment to the author(s)
* All users must log in before leaving a comment
Views 0
Downloads 0
Comments 0
Metrics 0


×
Alerts
Notify me about updates to this article or when a peer-reviewed version is published.
We use cookies on our website to ensure you get the best experience.
Read more about our cookies here.