On Stieltjes Polynomials and Gauss-kronrod Quadrature

Let D be a real function such that D(z) is analytic and D(z) ± 0 for \z\ < 1. Furthermore, put W(x) = \J\ x2\D(e'v)\2 , x = costp , tp e [0, 71 ], and denote by pn(', rV) the polynomial which is orthogonal on [-1, +1] to Pn_[ (P„_! denotes the set of polynomials of degree at most n 1 ) with respect to W . In this paper it is shown that for each sufficiently large n the polynomial En+X(-, W) (called the Stieltjes polynomial) of degree n + \ which is orthogonal on [-1,-1-1] to Pn with respect to the sign-changing function pn('> W)W has n + 1 simple zeros in (-1,1) and that the interpolation quadrature formula (called the Gauss-Kronrod quadrature formula) based on nodes which are the In + 1 zeros of En+l(-, W)pn(-, IV) has all weights positive.