Computation of Legendre series coefficients

Abstract

LEGSER approximates the first N + 1 coefficients B n of the Legendre series expansion of a function ƒ( x ) having known Chebyshev series coefficients A n . Several algorithms are available for the computation of coefficients A n of the truncated Chebyshev series expansion on [-1, 1] ƒ( x ) ≃ ∑′ N n =0 A n T n ( 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 B n of the expansion ƒ( x ) ≃ ∑ N n =0 B n P n ( x ) (2) is less suitable for practical use since it requires the abscissas and weights of the Gauss-Legendre quadrature formulas [2].