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Given a weight function @s(x) on [-1,1], or more generally a positive Borel measure, the Erdos-Turan theorem assures the convergence in L"2^@s-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 @s. 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 @w(@q)=@s(cos@q)|sin@q| on [-@p,@p]. 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. |
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