On the Filon and Levin methods for highly oscillatory integral

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 ∫ a b f ( x ) e i ω g ( x ) d x with the polynomial interpolation approach are identical when g ( x ) is a linear function. Based on this result, a new efficient Levin quadrature for ∫ a b f ( x ) e i ω g ( x ) d x is presented.