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Let there be given a probability measure µ on the unit circle T of the complex plane and consider the inner product induced by µ. In this paper we consider the problem of orthogonalizing a sequence of monomials {z r k } k , for a certain order of the r k ∈ Z, by means of the Gram-Schmidt orthogonalization process. This leads to a sequence of orthonormal Laurent polynomials {ψ k } k . We show that the matrix representation with respect to {ψ k } k of the operator of multiplication by z is an infinite unitary or isometric matrix allowing a 'snake-shaped' matrix factorization. Here the 'snake shape' of the factorization is to be understood in terms of its graphical representation via sequences of little line segments, following an earlier work of S. Delvaux and M. Van Barel. We show that the shape of the snake is determined by the order in which the monomials {z r k } k are orthogonalized, while the 'segments' of the snake are canonically determined in terms of the Schur parameters for µ. Isometric Hessenberg matrices and unitary five-diagonal matrices (CMV matrices) follow as a special case of the presented formalism.

In this paper we consider a general sequence of orthogonal Laurent polynomials on the unit circle and we first study the equivalences between recurrences for such families and Szegő's recursion and the structure of the matrix representation for the multiplication operator in Λ when a general sequence of orthogonal Laurent polynomials on the unit circle is considered. Secondly, we analyze the computation of the nodes of the Szegő quadrature formulas by using Hessenberg and five-diagonal matrices. Numerical examples concerning the family of Rogers-Szegő q-polynomials are also analyzed.

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Abstract. In this paper we characterize rational Szegő quadrature formulas associated with Chebyshev weight functions, by giving explicit expressions for the corresponding para-orthogonal rational functions and weights in the quadratures. As an application, we give characterizations for Szegő quadrature formulas associated with rational modifications of Chebyshev weight functions. Some numerical experiments are finally presented.

Abstract. In this paper, quadrature formulas on the real line with the highest degree of accuracy, with positive weights, and with one or two prescribed nodes anywhere on the interval of integration are characterized. As an application, the same kind of rules but with one or both (finite) endpoints being fixed nodes and one or two more prescribed nodes inside the interval of integration are derived. An efficient computation of such quadrature formulas is analyzed by considering certain modified Jacobi matrices. Some numerical experiments are finally presented.

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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.