On Quadrature Methods for Highly Oscillatory Integrals and Their Implementation

On highly oscillatory problems arising in electronic engineering

“…Note that we have implemented Filon-type methods in the most straightforward manner, with constant step size and without any error control (cf. [10] for error control for Filon-type methods). It is highly likely that more sophisticated implementation would have resulted in even more striking outcome.…”

“…The problem, though, is that the integral (1.3) does not fit into the framework of traditional asymptotic theory for highly oscillatory integrals [7,20]: the latter is concerned with integrals of the form Ω f (x)e iωg(x) dx, where ω 1 while neither f nor g are oscillatory. This is also the case with the methods for numerical calculation of highly oscillatory integrals that have been developed recently [6,10,16].…”

Finite-dimensional de Branges Subspaces Generated by Majorants

“…[4] proposes to cluster the interpolation points near the endpoints ±1. As in the case of the pure Filon quadrature, Assumption 1.2 is the key to ensure convergence as n → ∞:…”

“…Here, f and g are assumed to satisfy: Filon-type quadrature (see [4][5][6]) assumes that integrals 1 −1 e ikg(x) π(x) dx can be evaluated for polynomials π. Hence, a quadrature rule for the integral (1) can be obtained by replacing the integrand x → e ikg(x) f (x) with x → e ikg(x) I ∆ f (x), where I ∆ f is a polynomial (Hermite-)interpolant of f ; that is, we obtain the Filon-type quadrature rule…”

Asymptotic expansion and quadrature of composite highly oscillatory integrals

“…In the last few years this has been complemented by a comprehensive understanding of the numerical quadrature of such integrals by a range of methods: Filon-type [IN04,IN05], Levin-type [Lev96,Olv06], and numerical steepest descent [HV06]. It is however in the nature of mathematical research that, no sooner than we declare a theory 'complete', a new application comes to light, provides a counterexample, and challenges our understanding of the subject.…”