Averaging Sets on the Unit Circle Original Research Article

For every normalized measure σ on the unit circle T let tσ(n) be the maximal integer t such that the quadrature formula of Chebyshev type [formula] holds for some subset {(x1, y1),...(xn, yn)} of T and for all polynomials p(x, y) of deg p l≤ t. If ω is the Lebesgue measure then tω(n)= n − 1. Moreover, tσ(n) ≤ n − 1 for every σ. Under the Kolmogorov-Szegö condition on σ we prove that σ = ω if tσ(n) = n − 1 for a subsequence of n = 1, 2, 3,... .