Computation of Legendre series coefficients

LEGSER approximates the first N + 1 coefficients Bn of the Legendre series expansion of a function ƒ(x) having known Chebyshev series coefficients An. Several algorithms are available for the computation of coefficients An of the truncated Chebyshev series expansion on [-1, 1] ƒ(x) ≃ ∑′Nn=0 AnTn(x), (1) where ∑′ donotes a sum whose first term is halved. The commonly used algorithms are based on the orthogonal property of summation of the Chebyshev polynomials [1]. The application of the analogous property of the Legendre polynomials for the calculation of the coefficients Bn of the expansion ƒ(x) ≃ ∑Nn=0 BnPn(x) (2) is less suitable for practical use since it requires the abscissas and weights of the Gauss-Legendre quadrature formulas [2].

View this article in the ACM Digital Library.