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

This paper shows that for any suitably smooth function f(x)f(x) and arbitrarily selected interpolation nodes c1,c2,…,cvc1,c2,…,cv in [a,b][a,b], the Filon method and the Levin method for ∫abf(x)eiωg(x)dx with the polynomial interpolation approach are identical when g(x)g(x) is a linear function. Based on this result, a new efficient Levin quadrature for ∫abf(x)eiωg(x)dx is presented.