Quadrature identities for interlacing and orthogonal polynomials

Abstract:

Let $S$ be a real polynomial of degree $n$ with real simple zeros $\left \{ x_{j}\right \} _{j=1}^{n}$. Let $R$ be a real polynomial of degree $n-1$ whose zeros interlace those of $S$. We prove the quadrature identity \begin{equation*} \int _{-\infty }^{\infty }\frac {P(t)}{S^{2}\left ( t\right ) }h\left ( \frac {R}{S }\left ( t\right ) \right ) dt=\left ( \int _{-\infty }^{\infty }h\left ( t\right ) dt\right ) \sum _{j=1}^{n}\frac {P\left ( x_{j}\right ) }{\left ( RS^{\prime }\right ) \left ( x_{j}\right ) }, \end{equation*} valid for all polynomials $P$ of degree $\leq 2n-2$ and any $h\in L_{1}\left ( \mathbb {R}\right )$. We deduce identities involving orthogonal polynomials and weak convergence results involving orthogonal polynomials.