Highly oscillatory integrals of composite type arise in electronic engineering and their calculations is 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 of orthogonal polynomials and its nodes and weights can be computed efficiently by using tools of numerical linear algebra. We show that the rate of convergence of this rule depends solely on the regularity of the non-oscillatory part of the integrand. The second one is constructed with respect to a sign-changing function and the classical theory of Gaussian quadrature can not be used anymore. We explore theoretical properties of this Gaussian quadrature, including the trajectories of the quadrature nodes and the convergence rate of these nodes to the endpoints of the integration interval, and prove its asymptotic error estimate under suitable hypotheses. Numerical experiments are presented to demonstrate the performance of the proposed methods.

翻译：在电子工程中出现合成型的高度振动元件,其计算是一个具有挑战性的问题。在本文中,我们提出了用于计算此类元件的两种高斯二次曲线规则。第一种是建立在古典正统多面形理论及其节点和重量的基础上,可以通过使用数字线性代数工具来有效计算。我们表明,这一规则的趋同率完全取决于原体非振动部分的规律性。第二种是针对标志改变功能和高斯二次曲线经典理论而设计的。我们探索了高斯二次曲线的理论特性,包括四面形结点的轨迹和这些节点与整合间隔端点的趋同率,并证明在适当的假体下其无症状误差估计值。提出了数值实验,以展示拟议方法的性能。