On the numerical quadrature of highly-oscillating integrals I: Fourier transforms

Highly-oscillatory integrals are allegedly difficult to calculate.The main assertion of this paper is that that impression isincorrect. As long as appropriate quadrature methods are used,their accuracy increases when oscillation becomes faster andsuitable choice of quadrature points renders this welcome phenomenonmore pronounced. We focus our analysis on Filon-type quadratureand analyse its behaviour in a range of frequency regimes forintegrals of the form ₀^h f(x)e^{i x} w(x)d x, where h>0 issmall and | | large. Our analysis is applied to modified Magnus methods for highly-oscillatoryordinary differential equations. Received 6 June 2003. Revised 14 October 2003.