Estimates for the variance of positive quadrature formulas

We consider (positive) quadrature formulas Qn and their respective remainder terms Rn of algebraic degree deg(Rn) = m being defined in (1.1)–(1.3) below. By comparison with Gaussian formulas and with help of inequalities for Legendre polynomials, we prove new inequalities for the variance Var(Qn) of quadrature formulas Qn. In particular, for interpolatory quadrature formulas, as a consequence we have the following asymptotical bounds for the minimal variance: 4.14746limn→∞ < inf n Var (Qn < 4021763