On the order of indeterminate moment problems
arXiv:1310.0247 · doi:10.1016/j.aim.2013.09.020
Abstract
For an indeterminate moment problem we denote the orthonormal polynomials by P_n. We study the relation between the growth of the function P(z)=(\sum_{n=0}^\infty|P_n(z)|^2)^{1/2} and summability properties of the sequence (P_n(z)). Under certain assumptions on the recurrence coefficients from the three term recurrence relation zP_n(z)=b_nP_{n+1}(z)+a_nP_n(z)+b_{n-1}P_{n-1}(z), we show that the function P is of order αwith 0<α<1, if and only if the sequence (P_n(z)) is absolutely summable to any power greater than 2α. Furthermore, the order αis equal to the exponent of convergence of the sequence (b_n). Similar results are obtained for logarithmic order and for more general types of slow growth. To prove these results we introduce a concept of an order function and its dual. We also relate the order of P with the order of certain entire functions defined in terms of the moments or the leading coefficient of P_n
45 pages. To appear in Adv. Math