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In this paper we revisit some quadrature methods for highly oscillatoryintegrals of the form $\int_{-1}^1f(x)e^{{\rm i}\omega x}dx, \omega>0$.Exponentially Fitted (EF) rules depend on frequency dependent nodes which startoff at the Gauss-Legendre nodes when the frequency is zero and end up at theendpoints of the integral when the frequency tends to infinity. This makes therules well suited for small as well as for large frequencies. However, thecomputation of the EF nodes is expensive due to iteration and ill-conditioning.This issue can be resolved by making the connection with Filon-type rules. Byintroducing some $S$-shaped functions, we show how Gauss-type rules withfrequency dependent nodes can be constructed, which have an optimal asymptoticrate of decay of the error with increasing frequency and which are effectivealso for small or moderate frequencies. These frequency-dependent nodes canalso be included into Filon-Clenshaw-Curtis rules to form a class of methodswhich is particularly well suited to be implemented in an automatic softwarepackage.

Interpolatory quadrature rules for oscillatory integrals