Orthogonality, interpolation and quadratures on the unit circle and the interval

Given a weight function σ ( x ) on [ − 1 , 1 ] , or more generally a positive Borel measure, the Erdős–Turán theorem assures the convergence in L 2 σ -norm to a function f of its sequence of interpolating polynomials at the zeros of the orthogonal polynomials or equivalently at the nodes of the Gauss–Christoffel quadrature formulas associated with σ . In this paper we will extend this result to the nodes of the Gauss–Radau and Gauss–Lobatto quadrature formulas by passing to the unit circle and taking advantage of the results on interpolation by means of Laurent polynomials at the zeros of certain para-orthogonal polynomials with respect to the weight function ω ( θ ) = σ ( cos θ ) | sin θ | on [ − π , π ] . As a consequence, an application to the construction of certain product integration rules on finite intervals of the real line will be given. Several numerical experiments are finally carried out.