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Highly oscillatory quadrature: the story so far. (English) Zbl 1119.65317
Bermúdez de Castro, Alfredo (ed.) et al., Numerical mathematics and advanced applications. Proceedings of ENUMATH 2005, the 6th European conference on numerical mathematics and advanced applications, Santiago de Compostela, Spain, July 18--22, 2005. Berlin: Springer (ISBN 3-540-34287-7/hbk). 97-118 (2006).
Summary: The last few years have witnessed substantive developments in the computation of highly oscillatory integrals in one or more dimensions. The availability of new asymptotic expansions and a Stokes-type theorem allow for a comprehensive analysis of a number of old (although enhanced) and new quadrature techniques: the asymptotic, Filon-type and Levin-type methods. All these methods share the surprising property that their accuracy increases with growing oscillation. These developments are described in a unified fashion, taking the multivariate integral $\int_\Omega f(x)e^{i\omega g(x)}\,dV$ as our point of departure. For the entire collection see [Zbl 1103.65002].

##### MSC:
 65D32 Quadrature and cubature formulas (numerical methods) 41A55 Approximate quadratures 65T40 Trigonometric approximation and interpolation (numerical methods)