Quadrature Formulas on the Unit Circle and Two-Point Padé Approximation [chapter]

Adhemar Bultheel, Pablo González-Vera, Erik Hendriksen, Olav Njåstad
1994 Nonlinear Numerical Methods and Rational Approximation II  
In this paper our aim is to estimate integrals of the form [,"if} = J~... f(ei9)dl-£(8) where 1-£ is, in general a complex measure. We consider quadrature formulas like [n if} = L:7=1 Aj,nf(xj,n) with n distinct nodes Xj,n on the unit circle and so that [,"if} = [n{f} for any f E 'Rn (a certain subspace of Laurent polynomials with dimension n). Under appropriate assumptions on the function f we show that such quadratures are immediately related to the estimation of the so-called Riesz-Herglotz
more » ... ransform of the measure 1-£. Results concerning the rate of convergence of these quadratures are also given.
doi:10.1007/978-94-011-0970-3_26 fatcat:u4mw7suuorgf7iyn2uwqatm564