In this paper we deal with some perturbations of probability measures supported on the unit circle as well as, in a more general framework, with Hermitian linear functionals. We focus our attention in the Hessenberg matrix associated with the multiplication operator in terms of an orthogonal basis in the linear space of polynomials with complex coefficients. The LU and QR factorizations of such a matrix are introduced. Then, the connection between the above-mentioned perturbations and such factorizations is presented.
Abstract. In this paper we are concerned with the estimation of integrals on the unit circle of the form and weights {λ j } n j=1 in Szegö quadrature formulas are explicitly deduced. Illustrative numerical examples are also given.
We establish a relation between quadrature formulas on the interval [−1, 1], that approximate integrals of the form Jµ(F ) = 1 −1 F (x)µ(x)dx, and Szegő quadrature formulas on the unit circle that approximate integrals of the form Iω(f ) = π −π f e iθ ω(θ)dθ. The functions µ(x) and ω(θ) are assumed to be weight functions on [−1, 1] and [−π, π], respectively, and related by ω(θ) = µ(cos θ)| sin θ|. It is well known that the nodes of Szegő formulas are the zeros of the so called para-orthogonal polynomials Bn(z, τ ) = Φn(z) + τ Φ * n (z), |τ | = 1, Φn(z) and Φ * n (z) being the orthogonal and reciprocal polynomials, respectively, with respect to the weight function ω(θ). Furthermore, for τ = ±1 we have recently obtained Gauss-type quadrature formulas on [−1, 1], (see [1]). In this paper, making use of the para-orthogonal polynomials with τ = ±1, a one-parameter family of interpolatory quadrature formulas with positive coefficients for Jµ(F ) is obtained along with error expressions for analytic integrands. Finally, some illustrative numerical examples are also included.
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