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 E S（w）= （f： fxlf（x）lw（x）dx 〈 ∞ satisfying the conditions G（2J）（x） ：〉 O, x E （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 E S（w） with G ≥ 0 satisfying
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.
