A characterization of M. W. Wilson’s criterion for nonnegative expansions of orthogonal polynomials

Given a nonnegative function $f(x)$, M. W. Wilson observed that if \begin{equation}\int _0^\infty f(x) Q_i(x) Q_j(x)d\mu (x) \leqslant 0,\quad i \ne j, \quad \tag {$1$}\end{equation} then the polynomials ${P_n}(x),{P_n}(0) = 1$, orthogonal relative to $f(x)d\mu (x)$, have an expansion \[ {P_n}(x) = \sum \limits _{k = 0}^n {{a_{kn}}{Q_k}(x)} \] with nonnegative coefficients ${a_{kn}} \geqslant 0$ where ${Q_n}(x),{Q_n}(0) = 1$, are orthogonal relative to $d\mu (x)$. Recently it was shown that (1) holds for $f(x) = {x^c},0 < c < 1$. In this paper we characterize those functions $f(x)$ for which (1) is valid for all positive measures $d\mu (x)$.