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 impression is incorrect. As long as appropriate quadrature methods are used, their accuracy increases when oscillation becomes faster and suitable choice of quadrature points renders this welcome phenomenon more pronounced. We focus our analysis on Filon-type quadrature and analyse its behaviour in a range of frequency regimes for integrals of the form w(x)dx, where h > 0 is small and |#| large.

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