Legendre series analysis and computation via composed Abel-Fourier transform

Legendre coefficients of an integrable function f(x) are proved to coincide with the Fourier coefficients with a nonnegative index of a suitable Abel-type transform of the function itself. The numerical computation of N Legendre coefficients can thus be carried out efficiently in O(N log N) operations by means of a single fast Fourier transform of the Abel-type transform of f(x). Symmetries associated with the Abel-type transform are exploited to further reduce the computational complexity. The dual problem of calculating the sum of Legendre expansions at a prescribed set of points 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, some numerical tests are illustrated to exemplify and confirm the theoretical results.