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]=∫abf(x)cos(r1x)Jμ(rx)dx. For generalized Fourier transformation I[f]=∫abf(x)eiωg(x)dx under the condition that g′(x)≠0g′(x)≠0 for all x∈[a,b]x∈[a,b], Levin iteration method with the initial U[0](x)≡0U[0](x)≡0 is identical to the asymptotic method.