There is a nice necessary and sufficient criterion for a polynomial $p_n$ of degree $n$ with real coefficients to have all zeros real and simple. Namely, it says: $p_n$ has $n$ (distinct) real roots $x_{1},\dots,x_{n}$ iff the Hankel matrix $$S_{n}=\begin{pmatrix} s_{0} & s_{1} & \dots & s_{n-1}\\ s_{1} & s_{2} & \dots & s_{n}\\ \vdots & \vdots & \ddots & \vdots\\ s_{n-1} & s_{n} & \dots & s_{2n-2} \end{pmatrix}, \quad s_{k}:=\sum_{j=1}^{n}x_{j}^{k},$$ is positive definite; see, for example here.
My first question asks if there is some "infinite analogue" of the above equivalence where a polynomial is replaced by an entire function.
More concretely, let me assume the following. Let $f$ is a given entire function with real Taylor coefficients. Next, let $\{p_{n}\}_{n\geq1}$ be a sequence of polynomials (with increasing degree) approximating $f$. For example, we may assume $p_{n}\to f$, for $n\to\infty$, uniformly on compact subsets of $\mathbb{C}$. (One can take $p_n$ to be the truncated Taylor expansion of $f$, for instance.) Assume further that $x_{1}(n),\dots,x_{n}(n)$ are zeros of $p_n$ and that there exist limits $$\hskip100pt \lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n}\left(x_{j}(n)\right)^{k}=\alpha_{k},\hskip100pt (*)$$ for all $k\in\mathbb{N}$.
The second question is: if the Hankel matrix $A$, where $A_{i,j}:=\alpha_{i+j}$, $i,j\geq0$, is associated with a positive moment functional, i.e., all the principal sections of $A$ are positive definite, can one say something about zeros of $f$? Are they real and simple, for example?
The third question: Let me add one last related question. Assume that we have the sequence of polynomials $p_{n}$ as above, so the limits in $(*)$ hold, but we have no limit function $f$. The sequence $\{p_{n}\}$ is not necessarily convergent. Is it true that, if $A$ has all the principal sections positive definite, then the sequence $\{p_{n}\}$ is asymptotically real-rooted, that is, all the limit points of zeros are real?
A slight reformulation of the question as well as an addition of some other assumptions (for example, on the order of $f$, the rate of the convergence in the limits $(*)$, etc.) is possible.
