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 a nonnegative index suitable Abel-type transform itself. The numerical computation N can thus be carried out efficiently in O(NlogN) operations by means single fast f(x). Symmetries associated exploited further reduce computational complexity. dual problem calculating sum expansions at prescribed set points is also considered. We prove that series written as Abel series. This fact allows us state efficient algorithm for evaluation expansions. Finally, some tests illustrated exemplify and confirm theoretical results.