Bounds on order of indeterminate moment sequences. (English) Zbl 1375.44008
Fixing a sequence of real numbers, for which the associated Hamburger power moment problem is assumed to be solvable and indeterminate, the set of all representing measures of this sequence is parametrized with the help of the so-called Nevanlinna matrix, whose entries are entire functions of minimal exponential type, having the same order, called the order of the moment sequence. In 1939, M. S. Livšic found a lower estimate for the order of the moment sequence. In this paper, the authors show the existence of moment sequences whose order is strictly larger than the Livšic lower estimate. They also give an upper estimate, which has an explicit formula in terms of the canonical system associated with the given moment sequence. Adding a regularity assumption, they show that the upper estimate coincides with a lower estimate, allowing the computation of the order of the moment sequence.
MSC:
| 44A60 | Moment problems |
| 30D15 | Special classes of entire functions of one complex variable and growth estimates |
| 37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
| 47B36 | Jacobi (tridiagonal) operators (matrices) and generalizations |
| 34L20 | Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators |
Keywords:
indeterminate moment problem; canonical system; order of entire function; asymptotic of eigenvalues; Hamburger power moment problem; Nevanlinna matrixReferences:
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