Positivity of the weights of interpolatory quadrature formulae with Bernstein–Szegö abscissae

Abstract

We consider interpolatory quadrature formulae, relative to the Legendre weight function w(t) = 1 on [ − 1,1], having as nodes the zeros of any one of the nth degree orthogonal polynomials relative to the Bernstein–Szegö weight functions

$$w_{\gamma}^{\left(\pm 1/2\right)}(t)=\frac{\left(1-t^{2}\right)^{\pm 1/2}} {1-\displaystyle\frac{4\gamma}{(1+\gamma)^{2}}t^{2}},\ \ -1<t<1, \ \ -1<\gamma<1.$$

We derive semiexplicit formulae for the quadrature weights and show that the weights are almost all positive; exceptions occur only for the weight corresponding to the node at 0 when n is odd and are carefully identified.