The uniqueness of an orthogonality measure (Q2643086)

\textit{B. Wendroff} [Proc. Am. Math. Soc. 12, 554--555 (1961; Zbl 0099.05601)] proved that 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 \(z_{n+1,j}< z_{n,j}< z_{n+1,j+1}\) for \(j=1,\cdots,n\), the monic polynomials \(p_n(z)= \prod_{j=1}^n (z-z_{n,j})\) and \(p_{n+1}(z)= \prod_{j=1}^{n+1} (z-z_{n+1,j})\) can be embedded in an infinite sequence of monic orthogonal polynomials, which is not uniquely determined. In the present paper the authors prove, under the same hyphothesis, that there exists a uniquely determined probability measure \(\mu_0\), with \( \text{card}(\text{supp}(\mu_0))=n+1\), with respect to which \(p_n(z)= \prod_{j=1}^n (z-z_{n,j})\) and \(p_{n+1}(z)= \prod_{j=1}^{n+1} (z-z_{n+1,j})\) are orthogonal. The proof does not invoke Favard's theorem and Wendroff's theorem follows as an immediate consequence. They also consider sufficient conditions that ensure the uniqueness of the orthogonality measure arising in the context when the support of the measure has \(n+2\) real points.