Highly oscillatory integrals of composite type arise in electronic engineering and their calculations are a challenging problem. In this paper, we propose two Gaussian quadrature rules for computing such integrals. The first one is constructed based on the classical theory orthogonal polynomials its nodes weights can be computed efficiently by using tools numerical linear algebra. An interestin...

A straightforward 3-point quadrature formula of closed type is derived that improves on Simpson’s rule. Just using the additional information of the integrand’s derivative at the two endpoints we show the error is sixth order in grid spacing. Various error bounds for the quadrature formula are obtained to quantify more precisely the errors. Applications in numerical integration are given. With ...

In this paper, we consider smooth, real-valued random fields built up from i.i.d. copies of centered, unit variance smooth Gaussian fields on a manifold M . Specifically, we consider random fields of the form fp = F (y1(p), . . . , yk(p)) for F ∈ C(R;R) and (y1, . . . , yk) a vector of C centered, unit-variance Gaussian fields. For fields of this type, we compute the expected Euler characterist...