Monotony in interpolatory quadratures

Summary

Let$f(z) = \sum\limits_{j = 0}^\infty{t_{2j} z^{2j} } ,t_{2j}\geqq 0(j = 0,1,2,...)$, be holomorphic in an open disc with the centrez ₀=0 and radiusr>1. LetQ _n (n=1, 2, ...) be interpolatory quadrature formulas approximating the integral$\int\limits_{ - 1}^{ + 1} {f(x)dx}$. In this paper some classes of interpolatory quadratures are considered, which are based on the zeros of orthogonal polynomials corresponding to an even weight function. It is shown that the sequencesQ _n 9f] (n=1, 2, ...) are monotone. Especially we will prove monotony in Filippi's quadrature rule and with an additional assumption onf monotony in the Clenshaw-Curtis quadrature rule.