Convergence of modified approximants associated with orthogonal rational functions

AbstractLet {αn} be a sequence in the unit disk D = {z ∈ C: ¦z¦ < 1} consisting of a finite number of points cyclically repeated, and let L be the linear space generated by the functions Bn(z) = Πk=0n − αk(z − αk)¦αk¦(1 − αkz). Let {ϕn(z)} be orthogonal rational functions obtained from the sequence {Bn(z)} (orthogonalization with respect to a given functional on L), and let {ψn(z)} be the corresponding functions of the second kind (with superstar transforms ϕn∗(z) and ψn∗(z) respectively). Interpolation and convergence properties of the modified approximants Rn(z, un, vn) = (unψn(z) − vnψn∗(z))(unϕn(z) + vnϕn∗(z)) that satisfy ¦un¦ = ¦vn¦ are discussed

Quadrature formulas on the unit circle based on rational functions

AbstractQuadrature formulas on the unit circle were introduced by Jones in 1989. On the other hand, Bultheel also considered such quadratures by giving results concerning error and convergence. In other recent papers, a more general situation was studied by the authors involving orthogonal rational functions on the unit circle which generalize the well-known Szego&#x030B; polynomials. In this paper, these quadratures are again analyzed and results about convergence given. Furthermore, an application to the Poisson integral is also made