The uniqueness of an orthogonality measure

Given two finite sets of real points { z n , j } j = 1 n and { z n + 1 , j } j = 1 n + 1 satisfying the interlacing property z n + 1 , j < z n , j < z n + 1 , j + 1 , j = 1 , … , n , the monic polynomials p n ( z ) = ∏ j = 1 n ( z - z n , j ) and p n + 1 ( z ) = ∏ j = 1 n + 1 ( z - z n + 1 , j ) can be embedded in an infinite sequence of monic orthogonal polynomials (cf. [B. Wendroff, On orthogonal polynomials, Proc. Amer. Math. Soc. 12 (1961) 554–555. [6]). We discuss, in turn, the uniqueness of the measure of orthogonality μ arising in this context if card ( supp ( μ ) ) = n + 1 or n + 2 .