On the convergence rates of Legendre approximation

Abstract

Abstract. The problem of the rate of convergence of Legendre approximation is considered. We first establish the decay rates of the coefficients in the Legendre series expansion and then derive error bounds of the truncated Legendre series in the uniform norm. In addition, we consider Legendre approximation with interpolation. In particular, we are interested in the barycentric Lagrange formula at the Gauss-Legendre points. Explicit barycentric weights, in terms of Gauss-Legendre points and corresponding quadrature weights, are presented that allow a fast evaluation of the Legendre interpolation formula. Error estimates for Legendre interpolation polynomials are also given.

Iteration-Free Computation of Gauss--Legendre Quadrature Nodes and Weights

“…According to Theorem 3.1 in [6], this factor can be chosen such that the barycentric interpolation weights are given by:…”

Fast and Accurate Computation of Gauss--Legendre and Gauss--Jacobi Quadrature Nodes and Weights

“…where the constant C is arbitrary, the barycentric interpolation formula [5] p(x) = Thus if the x k are the roots of the Jacobi polynomial P (α,β) n (x), the derivative values P (α,β) n (x k ) needed to determine the corresponding barycentric weights can be computed in exactly the same way as for the quadrature weights [27,48].As such, we now have a fast, accurate, and stable [28] method of evaluating Jacobi interpolants, even at millions of points. Software MATLAB code for the Gauss-Legendre and Gauss-Jacobi algorithms described in this paper can be found in Chebfun's legpts and jacpts functions respectively [44].…”

“…However, there is some call for large global Legendre and Jacobi grids, for example in spectral methods and high-degree polynomial integration [44,50]. Furthermore, the relation between the quadrature and barycentric weights, as pointed out by Wang and Xiang [48,Theorem 3.1], allows the stable evaluation of Legendre and Jacobi interpolants.…”

An Introduction to Trajectory Optimization: How to Do Your Own Direct Collocation

“…Alternatively, the quadrature weights can be determined directly from the interpolation points and weights, although the equations are specific to each type of orthogonal polynomial. For example, the Legendre-Gauss quadrature weights and the Legendre-Gauss-Lobatto weights can be computed as More details on the calculation of quadrature rules can be found in [58,20,32,65]. Appendix E. Parameters for Example Problems.…”