From Asymptotics to Spectral Measures: Determinate Versus Indeterminate Moment Problems

From asymptotics to spectral measures: determinate versus indeterminate moment problems

In the field of orthogonal polynomials theory, the classical Markov theorem shows that for determinate moment problems the spectral measure is under control of the polynomials asymptotics. The situation is completely different for indeterminate moment problems, in which case the interesting spectral measures are to be constructed using Nevanlinna theory. Nevertheless it is interesting to observ...

To Determinate Moment Problems

It has been proved that algebraic polynomials P are dense in the space Lp(R, dμ), p∈ (0,∞), iff the measure μ is representable as dμ = wp dν with a finite non-negative Borel measure ν and an upper semi-continuous function w : R → R+ : = [0,∞) such that P is a dense subset of the space C0 w : = {f ∈ C(R) : w(x)f(x) → 0 as |x| → ∞} equipped with the seminorm ‖f‖w := supx∈R w(x)|f(x)|. The similar...

From Discrete to Absolutely Continuous Solutions of Indeterminate Moment Problems

We consider well-known families of discrete solutions to indeterminate moment problems and show how they can be used in a simple way to generate absolutely continuous solutions to the same moment problems. The following cases are considered: log-normal, generalized Stieltjes-Wigert, q-Laguerre and discrete q-Hermite II.

On the order of indeterminate moment problems

For an indeterminate moment problem we denote the orthonormal polynomials by Pn. We study the relation between the growth of the function P (z) = ( ∑∞ n=0 |Pn(z)|) and summability properties of the sequence (Pn(z)). Under certain assumptions on the recurrence coefficients from the three term recurrence relation zPn(z) = bnPn+1(z) + anPn(z) + bn−1Pn−1(z), we show that the function P is of order ...

On in nitely divisible solutions to indeterminate moment problems