Szegő–Lobatto quadrature rules

Abstract

Gauss-type quadrature rules with one or two prescribed nodes are well known and are commonly referred to as Gauss–Radau and Gauss–Lobatto quadrature rules, respectively. Efficient algorithms are available for their computation. Szegő quadrature rules are analogs of Gauss quadrature rules for the integration of periodic functions; they integrate exactly trigonometric polynomials of as high degree as possible. Szegő quadrature rules have a free parameter, which can be used to prescribe one node. This paper discusses an analog of Gauss–Lobatto rules, i.e., Szegő quadrature rules with two prescribed nodes. We refer to these rules as Szegő–Lobatto rules. Their properties as well as numerical methods for their computation are discussed.