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Asymptotic properties of Stieltjes polynomials and Gauss-Kronrod quadrature formulae. (English) Zbl 0828.41019
Summary: Stieltjes polynomials are orthogonal polynomials with respect to the sign changing weight function $$wP_n (\cdot, w)$$, where $$P_n (\cdot, w)$$ is the $$n$$th orthogonal polynomial with respect to $$w$$. Zeros of Stieltjes polynomials are nodes of Gauss-Kronrod quadrature formulae, which are basic for the most frequently and quadrature routines with combined practical error estimate. For the ultraspherical weight function $$w_\lambda (x)= (1- x^2 )^{\lambda- 1/2}$$, $$0\leq \lambda\leq 1$$, we prove asymptotic representations of the Stieltjes polynomials and of their first derivative, which hold uniformly for $$x= \cos \theta$$, $$\varepsilon\leq \theta\leq \pi-\varepsilon$$, where $$\varepsilon\in (0, \pi/2)$$ is fixed. Some conclusions are made with respect to the distribution of the zeros of Stieltjes polynomials, proving an open problem of G. Monegato [Numerische Integration, Tag. Oberwolfach 1978, ISNM 45, 231-240 (1979; Zbl 0416.65017)] and F. Peherstorfer [J. Approx. Theory 70, No. 2, 156-190 (1992; Zbl 0760.41020)]. As a further application, we prove an asymptotic representation of the weights of Gauss-Kronrod quadrature formulae with respect to $$w_\lambda$$, $$0\leq \lambda\leq 1$$, and we prove the precise asymptotical value for the variance of Gauss-Kronrod quadrature formulae in these cases.