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R. Pruckner, R. Romanov, H. Woracek:
"Bounds on order of indeterminate moment sequences";
in: "ASC Report 38/2015", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2015, ISBN: 978-3-902627-08-7, 1 - 26.

We investigate the order ρ of the four entire functions in the Nevanlinna matrix of an indeterminate Hamburger moment sequence. In operator theoretic language, this is the asymptotic behaviour of the eigenvalues of the associated Jacobi operator. We give upper and lower estimates for ρ which are explicit in terms of the parameters of the canonical system associated with the moment sequence via its three-term recurrance. Under a weak regularity assumption these estimates coincide, and hence ρ becomes computable. Dropping regularity leads to examples where the bounds do not coincide and do not coincide with the order. In particular this provides examples that in an estimate for order due to M.S.Livˇsic in 1939 equality does not always hold. Our proofs proceed via the theory of canonical systems.

: Indeterminate moment problem, canonical system, order of entire function, asymptotic of eigenvalues

http://www.asc.tuwien.ac.at/preprint/2015/asc38x2015.pdf