On the Levin iterative method for oscillatory integrals

This paper considers and gives error analysis for Levin iteration method to approximate Bessel-trigonometric transformation I [ f ] = ∫ a b f ( x ) cos ( r 1 x ) J μ ( rx ) d x . For generalized Fourier transformation I [ f ] = ∫ a b f ( x ) e i ω g ( x ) d x under the condition that g ′ ( x ) ≠ 0 for all x ∈ [ a , b ] , Levin iteration method with the initial U [ 0 ] ( x ) ≡ 0 is identical to the asymptotic method.