A lower bound for orthogonal polynomials with an application to polynomial hypergroups
This paper presents the following result: Let \((P_n )_{n\geq 0}\) be a sequence of orthonormal polynomials with respect to some measure \(\mu\) on \(\mathbb{R}\). Assume that the coefficients \(\lambda_n\) in the recurrence relation \[ x P_n (x)= \lambda_n P_{n+1} (x)+ \beta_n P_n (x)+ \lambda_{n-1} P_{n-1} (x) \] are bounded. Then, for all \(z\in \mathbb{C}\setminus \text{supp } \mu\), \[ \liminf_{n\to\infty} \root n\of {|P_n (z)|^2+ |P_{n+1} (z) |^2} >1. \] Applied to polynomial hypergroups, it is also shown that, under certain restrictions, the dual space of such a hypergroup is equal to \(\text{supp } \mu\). This supplements results of \textit{M. Vogel} [Math. Z. 194, 99-116 (1987; Zbl 0609.43004)] and the reviewer [Math. Z. 198, No. 3, 405-421 (1988; Zbl 0677.43002)] concerning the support of Plancherel measure of commutative hypergroups. The results there are obtained by completely different methods due to Hulanicki.
scientific article; zbMATH DE number 756649