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.

We consider a Clenshaw–Curtis–Filon-type method for highly oscillatory Bessel transforms. It is based on a special Hermite interpolation polynomial at the Clenshaw–Curtis points that can be efficiently evaluated using $O(NlogN)$ operations, where N is the number of Clenshaw–Curtis points in the interval of integration. Moreover, we derive corresponding error bounds in terms of the frequency and the approximating polynomial. We then show that this method yields an efficient numerical approximation scheme for a class of Volterra integral equations containing highly oscillatory Bessel kernels (a problem for which standard numerical methods fail), and it also allows the study of the asymptotics of the solutions. Numerical examples are used to illustrate the efficiency and accuracy of the Clenshaw–Curtis–Filon-type method for approximating these highly oscillatory integrals and integral equations.

We define a new fundamental constant associated with a P-matrix and show that this constant has various useful properties for the P-matrix linear complementarity problems (LCP). In particular, this constant is sharper than the Mathias–Pang constant in deriving perturbation bounds for the P-matrix LCP. Moreover, this new constant defines a measure of sensitivity of the solution of the P-matrix LCP. We examine how perturbations in the data affect the solution of the LCP and efficiency of Newton-type methods for solving the LCP.