Quadrature identities for interlacing and orthogonal polynomials

Abstract

Let S S be a real polynomial of degree n n with real simple zeros { x j } j = 1 n \left \{ x_{j}\right \} _{j=1}^{n} . Let R R be a real polynomial of degree n − 1 n-1 whose zeros interlace those of S S . We prove the quadrature identity ∫ − ∞ ∞ P ( t ) S 2 ( t ) h ( R S ( t ) ) d t = ( ∫ − ∞ ∞ h ( t ) d t ) ∑ j = 1 n P ( x j ) ( R S ′ ) ( x j ) , \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 P of degree ≤ 2 n − 2 \leq 2n-2 and any h ∈ L 1 ( R ) h\in L_{1}\left ( \mathbb {R}\right ) . We deduce identities involving orthogonal polynomials and weak convergence results involving orthogonal polynomials.