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：复合型高度振荡积分在电子工程中出现，其计算是一个具有挑战性的问题。本文提出了两种用于计算此类积分的高斯求积规则。第一种基于经典正交多项式理论构建，其节点和权值可利用数值线性代数工具高效计算。我们证明该规则的收敛速度仅取决于被积函数非振荡部分的正则性。第二种基于符号变化函数构建，此时经典高斯求积理论不再适用。我们探究了该高斯求积的理论性质，包括求积节点的轨迹、这些节点向积分区间端点收敛的速率，并在适当假设下证明了其渐近误差估计。数值实验展示了所提方法的性能。