On the asymptotic behaviour of functions of the second kind and Stieltjes polynomials and on the Gauss-Kronrod quadrature formulas

Abstract

First we study the asymptotic behaviour on the unit circle of functions of the second kind associated with polynomials orthogonal on the unit circumference. With the help of these results we derive, as in the case of orthogonal polynomials, the asymptotic behaviour of functions of the second kind associated with polynomials orthogonal on the interval [−1, 1]. Special attention is given to the asymptotic behaviour on the interval [−1, 1]. Using the known close connection between the Stieltjes polynomials and the functions of the second kind we find that the Stieltjes polynomial E n + 1 (·, (1 − x 2) w) is asymptotically equal to the orthogonal polynomial p n+1 ( x, w), if w( x)√1 − x 2 is positive and twice continuously differentiable on [−1, 1]. Furthermore we give, for sufficiently large n, several “interlacing properties” for the zeros of the Stieltjes polynomials, such as the interlacing property of the zeros of two consecutive Stieltjes polynomials, of the zeros of E n + 1 (·, (1 − x 2) w) and P n (·, w), etc. Finally we show that for sufficiently large n the Gauss-Kronrod quadrature formula has all quadrature weights positive, if the weight function satisfies the abovementioned conditions.