A product quadrature algorithm by Hermite interpolation

摘要

This paper is concerned with numerical integration of ∫1−1f(x)k(x)dx by product integration rules based on Hermite interpolation. Special attention is given to the kernel k(x) = eiτx, with a view to providing high precision rules for oscillatory integrals. Convergence results and error estimates are obtained in the case where the points of integration are zeros of pn(W; x) or of (1 − x2)pn−2(W; x), where pn(W; x), n = 0, 1, 2…, are the orthonormal polynomials associated with a generalized Jacobi weight W. Further, examples are given that test the performance of the algorithm for oscillatory weight functions.