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The main objective is to generalize previous results obtained for orthogonal Laurent polynomials and their application in the context of Stieltjes moment problems to the multipoint case. The measure of orthogonality is supposed to have support on [0, infinity) while the orthogonal rational functions will have poles that are assumed to be "in the neighborhood of 0 and infinity". In this way orthogonal Laurent polynomials will be a special case obtained when all the poles are at 0 and infinity. We shall introduce the restrictions on the measure and the locations of the poles gradually and derive recurrence relations, Christoffel-Darboux relations, and the solution of the rational Stieltjes moment problem under appropriate conditions.

We generalize a result by L. Golinskii¹ on separation of the zeros of para-orthogonal polynomials on the unit circle to a similar result for para-orthogonal rational functions. ¹ L. Golinskii, Quadrature formula and zeros of para-orthogonal polynomials on the unit circle, Acta Math. Hungar. 96 169-186, 2002. ispartof: Revista de la Academia Canaria de Ciencias vol:16 issue:1 pages:9-14 status: published

The main objective is to generalize previous results obtained for orthogonal Laurent polynomials and their application in the context of Stieltjes moment problems to the multipoint case. The measure of orthogonality is supposed to have support on [0,∞) while the orthogonal rational functions will have poles that are assumed to be “in the neighborhood of 0 and ∞”. In this way orthogonal Laurent polynomials will be a special case obtained when all the poles are at 0 and ∞. We shall introduce the restrictions on the measure and the locations of the poles gradually and derive recurrence relations, Christoffel–Darboux relations, and the solution of the rational Stieltjes moment problem under appropriate conditions.

From the Erdös-Turán theorem, it is known that if f is a continuous function on T={z:|z|=1} and L_n(f,z) denotes the unique Laurent polynomial interpolating f at the (2n+1)th roots of unity, then lim_{n→∞} ∫_T |f(z)-L_n(f,z)|^2| |d z|=0. Several years later, Walsh and Sharma gave a similar result but now considering a function analytic in D={z:|z