The index of determinacy for measures and the $l^ 2$-norm of orthonormal polynomials

Abstract:

For determinate measures $\mu$ having moments of every order we define and study an index of determinacy which checks the stability of determinacy under multiplication by even powers of $|t - z|$ for $z$ a complex number. Using this index of determinacy, we solve the problem of determining for which $z \in \mathbb {C}$ the sequence ${(p_n^{(m)}(z))_n}(m \in \mathbb {N})$ belongs to ${\ell ^2}$, where ${({p_n})_n}$ is the sequence of orthonormal polynomials associated with the measure $\mu$.