instituto de matemáticas universidad de sevilla
Antonio de Castro Brzezicki
Indeterminate Hamburger Moment Problems
Actividades de divulgación
A Hamburger moment sequence $\ (s_n) $  is a sequence of the form
$\ s_{n}=\int_{-\infty }^{\infty }x^{n}d\mu (x), n=0,1,..., $         (*)
Where $\ \mu $ is a positive measure on $\ \mathbb{R} $. It can be characterized by positive semidefiniteness of the infinite Hankel matrix $\ H={s_{m+n}} $.
In case $\ \mu $ is a probability measure with infinite support, the orthonormal polynomials $\ P_{n} , n\geqslant 0 $ with respect to $\ \mu $ (i.e. $\ \int P_{n}P_{m}d\mu=\delta _{n,m} $) satisfy a three-term recurrence relation
$\ xP_{n}(x)=b_{n}P_{n+1}(x)+b_{n-1}P_{n-1}(x),\;\; \; \; \; \; n\geq 0 $        (∗∗)
with real sequences $\ (a_n),(b_n) $ such that $\ b_n> 0 $.
Favard’s Theorem states that given $\ (a_n),(b_n) $ with $\ b_n> 0 $, then (∗∗) and the initial conditions $\ P_{-1}(x)=0,P_0(x)=1 $ determine a sequence of polynomials $\ P_n $, which are orthonormal with respect to a probability measure $\ \mu $ with infinite support.
Given $\ (a_n),(b_n) $ the moment problem consists in finding $\ \mu $ such that (∗) and (∗∗) hold.
The moment problem can be determinate or indeterminate depending if there is exactly one or several measures \mu on the real line such that these equations hold.
The indeterminate case is characterized by the convergence of the series $\ \sum \left | P_n(z) \right |^2 $ for all complex $\ \textit{z} $, and it leads to a reproducing kernel Hilbert space $\ \varepsilon $ of entire functions with the reproducing kernel
$\ \mathrm{K}(z,w)=\sum_{n=0}^{\infty}P_n(z)P_n(w)=\sum_{k,l=0}^{\infty}a_{k,l}z^kw^l,\; \; \; z,w\in \mathbb{C} $
In the indeterminate case the complete description of all solutions to the moment problem depends on four entire functions $\ \textit{A,B,C,D} $ of common order and type called the order and type of the moment problem.
The function $\ \mathrm{D} $ is given as
$\ \mathrm{D}(z)=zK(z,0)=z\sum_{n=0}^{\infty}P_n(z)P_n(0),\; \; \; z\in \mathbb{C} $
but in general it is difficult to calculate $\ \mathrm{D} $. It is therefore of some interest to be able to calculate the order and type of the moment problem directly from the coefficients $\ (a_n),(b_n) $ without calculating first $\ (P_n) $ and $\ \mathrm{D} $. This has been achieved in joint work with Ryszard Szwarc for certain classes of sequences $\ (a_n),(b_n) $.
In the indeterminate case we can consider the infinite matrix $\ \textit{A}=\left \{ a_{k,l} \right \} $ of coefficients in the kernel $\ \mathrm{K} (z, w) $. We shall see that in certain indeterminate cases— but not all—the following infinite matrix equations hold: $\ \textit{AH=HA=I} $.
I shall expose some of this classical theory and report on some recent results, which are joint work with Ryszard Szwarc