A note on asymptotic zero distribution of orthogonal polynomials.
Let \(a\in\mathbb{R}_+\) and \(b\in\mathbb{R}\) be constants, and consider a sequence of orthogonal polynomials given as \(q_{-1}(x)\equiv 0\), \(q_0(x)= 1\), and \[ aq_{n+1}(x)+ (b- x)q_n(x)+ aq_{n-1}(x)\equiv 0,\quad n\geq 1, \] with zeros \(y_{1n}< y_{2n}<\cdots< y_{nn}\) for \(q_n\). Together with the zeros the author considers the eigenvalues \(x_{1n}, x_{2n},\dots, x_{nn}\) of a symmetric \(n\times n\) matrix \((c_{ij})\) that satisfies (i) \(c_{ij}= 0\) unless \(| i- j|\) equals 0 or 1, and (ii) there are -- in a sense made precise in the paper -- not too many large differences \(| a- c_{i,i\pm 1}|\) and \(| b- c_{ii}|\). Then, the author's main result may be stated as follows \[ \sum^\infty_{r=1}| F(x_{rn})- F)y_{rn})|= o(n) \] for all real functions \(F\) that are bounded and continuous on some open interval \(I\supseteq [b- 2a, b+ 2a]\).
scientific article; zbMATH DE number 2013058