On the Filon and Levin methods for highly oscillatory integral ∫ab f(x)eiωg(x) dx

This paper shows that for any suitably smooth function f(x) and arbitrarily selected interpolation nodes c(1), c(2),..., c(v), in [a, b], the Filon method and the Levin method for integral(b)(a) f(x)e(i omega g(x)) dx with the polynomial interpolation approach are identical when g(x) is a linear function. Based on this result, a new efficient Levin quadrature for integral(a)(b) f(x)e(i omega g(x)) dx is presented. (C) 2006 Published by Elsevier B.V.