Leyla Daruis

A connection between quadrature formulas on the unit circle and the interval [ - 1,1]

Abstract We establish a relation between Gauss quadrature formulas on the interval [−1,1] that approximate integrals of the form I σ (F)= ∫ −1 +1 F(x)σ(x) d x and Szegő quadrature formulas on the unit circle of the complex plane that approximate integrals of the form I ω (f)= ∫ − π π f( e i θ )ω(θ) d θ . The weight σ(x) is positive on [−1,1] while the weight ω(θ) is positive on [−π,π]. It is shown that if ω(θ)=σ( cos θ)| sin θ| , then there is an intimate relation between the Gauss and Szegő quadrature formulas. Moreover, as a side result we also obtain an easy derivation for relations between orthogonal polynomials with respect to σ(x) and orthogonal Szegő polynomials with respect to ω(θ). Inclusion of Gauss–Lobatto and Gauss–Radau formulas is natural.

Szegö quadrature formulas for certain Jacobi-type weight functions

In this paper we are concerned with the estimation of integrals on the unit circle of the form ∫02π f(eiθ)ω(θ)dθ by means of the so-called Szego quadrature formulas, i.e., formulas of the type Σj=1n λjf(xj) with distinct nodes on the unit circle, exactly integrating Laurent polynomials in subspaces of dimension as high as possible. When considering certain weight functions ω(θ) related to the Jacobi functions for the interval [-1, 1], nodes {xj}j=1n and weights {λj}j=1n in Szego quadrature formulas are explicitly deduced. Illustrative numerical examples are also given.

Szegö polynomials and quadrature formulas on the unit circle

In this paper, computation of the so-called Szego quadrature formulas is considered for some special weight functions on the unit circle. The case of the Lebesgue measure is more deeply analyzed. Illustrative numerical examples are also given.

Interpolatory quadrature formulas on the unit circle for Chebyshev weight functions

Summary. In this paper, interpolatory quadrature formulas based upon the roots of unity are studied for certain weight functions. Positivity of the coefficients in these formulas is deduced along with computable error estimations for analytic integrands. A comparison is made with Szegö quadrature formulas. Finally, an application to the interval [-1,1] is also carried out.

A note on hermite-fejér interpolation for the unit circle

In this note, an extension to the unit circle of the classical Hermite-Fejer Theorem is given.

Gaussian quadrature formulae on the unit circle

Let µ be a probability measure on [0, 2π]. In this paper we shall be concerned with the estimation of integrals of the form Iµ(f)= (1/2π) ∫02π f(eiθ)dµ(θ). For this purpose we will construct quadrature formulae which are exact in a certain linear subspace of Laurent polynomials. The zeros of Szego polynomials are chosen as nodes of the corresponding quadratures. We will study this quadrature formula in terms of error expressions and convergence, as well as, its relation with certain two-point Pade approximants for the Herglotz-Riesz transform of µ. Furthermore, a comparison with the so-called Szego quadrature formulae is presented through some illustrative numerical examples.