In classical theorems on the convergence of Gaussian quadrature formulas for power orthogonal polynomials with respect to a weight w on I =(a,b),a function G ∈ S(w):= { f:∫_I | f(x)| w(x)d x < ∞} satisfying the conditions G ^(2j)(x) ≥ 0,x ∈(a,b),j = 0,1,...,and growing as fast as possible as x → a + and x → b,plays an important role.But to find such a function G is often difficult and complicated.This implies that to prove convergence of Gaussian quadrature formulas,it is enough to find a function G ∈ S(w) with G ≥ 0 satisfying sup n ∑λ0knG(xkn) k=1 n<∞ instead,where the xkn 's are the zeros of the n th power orthogonal polynomial with respect to the weight w and λ0kn 's are the corresponding Cotes numbers.Furthermore,some results of the convergence for Gaussian quadrature formulas involving the above condition are given.
A uniqueness theorem of a solution of a system of nonlinear equations is given. Using this result uniqueness theorems for power orthogonal polynomials, for a Gaussian quadrature formula of an extended Chebyshev system, and for a Gaussian Birkhoff quadrature formula are easily deduced.
