On Quadrature of Highly Oscillatory Functions (Journal of Computational Mathematics, 2006)
Some quadrature methods for integration of $\int_a^b f(x)e^{i ω g(x)}dx$ for rapidly oscillatory functions are presented. These methods, based on the lower order remainders of Taylor expansion and followed the thoughts of Stetter [9], Iserles and Nørsett [5], are suitable for all $ω$ and the decay of the error can be increased arbitrarily in case that $g'(x) ot=0$ for $x\in [a,b]$, and easy to be implemented and extended to the improper integration and the general case $ I[f]=\int_a^b f(x)e^{ig(ω,x)} dx$.
Keywords: Oscillatory integral; quadrature; Filon-type method; Taylor expansion.