Zeros of linear combinations of orthogonal polynomials
Antonio J. Durán
TL;DR
The paper proves that for any positive measure $\mu$ on $\mathbb{R}$, any fixed $K$ and any real coefficients $\gamma_0=1,\dots,\gamma_K$ with $\gamma_K\neq 0$, there exists a normalization sequence $\rho_n$ such that the finite linear combinations $q_n(x)=\sum_{j=0}^K\gamma_j p_{n-j}(x)$ have all real, simple zeros for all sufficiently large $n$, and these zeros interlace those of $p_{n-1}$. The authors provide constructive bounds on $\rho_n$ and the threshold $n_0$, and establish similar interlacing results for the nontrivial classical families (Hermite, Laguerre, Jacobi) with explicit Mehler–Heine-type asymptotics for the zeros. The method hinges on representing $q_n$ as $q_n(x)=A(x)p_n(x)+B(x)p_{n-1}(x)$, bounding $A$ and $B$, and leveraging Obreshkov’s interlacing theorem. These results extend Shohat’s early observations by giving a robust, renormalization-based framework to guarantee real-rootedness and interlacing for large degrees, with concrete implications for zero-distribution and quadrature-related properties.
Abstract
Given a sequence of orthogonal polynomials $(p_n)_n$ with respect to a positive measure in the real line, we study the real zeros of finite combinations of $K+1$ consecutive orthogonal polynomials of the form $$ q_n(x)=\sum_{j=0}^Kγ_jp_{n-j}(x),\quad n\ge K, $$ where $γ_j$, $j=0,\cdots ,K$, are real numbers with $γ_0=1$, $γ_K\not =0$ (which do not depend on $n$). We prove that for every positive measure $μ$ there always exists a sequence of orthogonal polynomials with respect to $μ$ such that all the zeros of the polynomial $q_n$ above are real and simple for $n\ge n_0$, where $n_0$ is a positive integer depending on $K$ and the $γ_j$'s.