A shortcut to asymptotics for orthogonal polynomials
Let \(q_ k\) be the sequence of orthonormal polynomials with respect to the positive measure \(\mu\) with real support \(E\). These satisfy the well known three term recurrence relation \(zq_ n(z)= \beta_ n q_{n+1} (z)+ \alpha_ n q_ n(z)+ \beta_{n-1} q_{n-1}(z)\). The Nevai class \(M(0,1)\) is the class of measures for which \(\alpha_ n\) tends to 0 and \(\beta_ n\) to \(1\over 2\). Nevai (1979) showed that for \(\mu\in M(0,1)\), the support \(E\) consists of \([-1,1]\) and at most a denumerable set of points outside this interval. Moreover, if \(\Phi\) is the conformal map of \(\mathbb{C}\setminus [-1,1]\) onto \(| z|>1\), then \(q_{n+1} (z)/ q_ n(z)\) tends to \(\Phi(z)\) locally uniformly in \(\mathbb{C} \setminus E\) and to \(1/ \Phi(z)\) pointwise in \(E\setminus [-1,1]\). In this paper new proofs for these results and related ones are provided. They rely on writing the recurrence relation as a 2 by 2 system of first order recurrences. The eigenvalues of the limit of the recurrence matrix are precisely \(\Phi(z)\) and \(1/\Phi(z)\). The ratio asymptotics thus correspond to the convergence to eigenvalues and this is related to general theorems for recurrence relations, the power method for eigenvalue computation and Poincaré's theorem. An historical survey about these topics and preliminary results about the zero distribution and the normal family properties of the ratio's are also included. In the last section, convergence results for diagonal Padé approximants to Hamburger functions are derived. The ratio asymptotics allow to accelerate this sequence of approximants by the \(\Delta^ 2\) method. This is also proved.
scientific article; zbMATH DE number 643755