Multiple orthogonality and applications in numerical integration

Abstract

In this paper, a brief survey of multiple orthogonal polynomials defined using orthogonality conditions spread out over r different measures are given. We consider multiple orthogonal polynomials on the real line, as well as on the unit semicircle in the complex plane. Such polynomials satisfy a linear recurrence relation of order r + 1, which is a generalization of the well known three-term recurrence relation for ordinary orthogonal polynomials (the case r = 1). A method for the numerical construction of multiple orthogonal polynomials by using the discretized Stieltjes-Gautschi procedure are presented. Also, some applications of such orthogonal systems to numerical integration are given. A numerical example is included.