A fast and simple algorithm for the computation of Legendre coefficients

Abstract

We present an ${\mathcal{O}({N\,{\rm log}\,N})}$ algorithm for the calculation of the first N coefficients in an expansion of an analytic function in Legendre polynomials. In essence, the algorithm consists of an integration of a suitably weighted function along an ellipse, a task which can be accomplished with Fast Fourier Transform, followed by some post-processing. The mathematical underpinning of this algorithm is an old formula that expresses expansion coefficients ${\hat{f}_m}$ as infinite linear combinations of derivatives. We evaluate the latter with the Cauchy theorem, thereby expressing each ${\hat{f}_m}$ as a scaled integral of ${f(z)\varphi_m(z)/z^{m+1}}$ along an appropriate contour, where ${\varphi_m}$ is a slowly converging hypergeometric function. Next, we transform ${\varphi_m}$ into another hypergeometric function which converges rapidly. Once we replace the latter function by its truncated Taylor expansion and choose an appropriate elliptic contour, we obtain an expression for the ${\hat{f}_m}$s which is amenable to rapid computation.