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Motivated by the works of Genin (1988, 1991), who studied paraorthogonal polynomials associated with positive definite Hermitian linear functionals and their corresponding recurrence relations, we provide paraorthogonality theory, in the context of quasidefinite Hermitian linear functionals, with a recurrence relation and the analogous result to the classical Favard's theorem or spectral theorem. As an application of our results, we prove that for any two monic polynomials whose zeros are simple and strictly interlacing on the unit circle, with the possible exception of one of them which could be common, there exists a sequence of paraorthogonal polynomials such that these polynomials belong to it. Furthermore, an application to the computation of Szegő quadrature formulas is also discussed.

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We present a relation between rational Gauss-type quadrature formulas that approximate integrals of the form (x), and rational Szegő quadrature formulas that approximate integrals of the formThe measures µ andμ are assumed to be positive bounded Borel measures on the interval [−1, 1] and the complex unit circle respectively, and are related byμ ′ (θ) = µ ′ (cos θ) |sin θ|. Next, making use of the so-called para-orthogonal rational functions, we obtain a one-parameter family of rational interpolatory quadrature formulas with positive weights for J µ (F). Finally, we include some illustrative numerical examples.

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In this paper, a new approach in the estimation of weighted integrals of periodic functions on unbounded intervals of the real line is presented by considering an associated weight function on the unit circle and making use of both Szegő and interpolatory type quadrature formulas. Upper bounds for the estimation of the error are considered along with some examples and applications related to the Rogers-Szegő polynomials, the evaluation of the Weierstrass operator, the Poisson kernel and certain strong Stieltjes weight functions. Several numerical experiments are finally carried out. Dedicated to Guillermo López-Lagomasino on the occasion of his 60th birthday.

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a b s t r a c tAs a continuation of the well known connection between the theory of orthogonal polynomials on the unit circle and the interval [−1, 1], in this paper properties concerning error and convergence of certain rational approximants associated with the measures dµ(t) and dσ (θ ) = |dµ(cos θ )| supported on [−1, 1] and the unit circle respectively are deduced. Numerical illustrations are also given.