On fast algorithms for the evaluation of Legendre coefficients

In this paper, we present formulas betwixt Legender and Chebyshev expansion coefficients for a piecewise smooth (or Dini–Lipschitz) function, analyze the error bounds for the Piessens’ algorithm and present a new algorithm with O(NlogN)O(NlogN) operations for computation of the first N+1N+1 coefficients of the Legendre expansion. Finally, we show the identity to the formulas given by Iserles [5] for analytic functions in a neighborhood of [−1,1][−1,1].