Can we characterize the space of functions which is real analytic but not real entire?

Question

A complex valued function $F,$ defined on an open set $E$ in the plane $\mathbb R^{2}$, is said to be real-analytic in $E$ if to every point $(s_{0}, t_{0})$ in there corresponds an expansion with complex coefficients $$F(s, t)= \sum_{n,m=0}^{\infty} a_{nm}(s-s_{0})^{m} (t-t_{0})^{n},$$ which converges absolutely for all $(s,t)$ in some neighbourhood of $(s_{0}, t_{0}).$

If $F$ is defined in the whole plane $\mathbb R^{2}$ by a series $$F(s, t)= \sum_{n,m=0}^{\infty} a_{nm}s^{m} t^{n},$$ which converges absolutely for every $(s,t),$ the we call $F$ real-entire.

Let us introduce temporary notations, $$RA(\mathbb R^{2}):=\text{The space of real analytic functions on $\mathbb R^{2}$},$$ and $$RE(\mathbb R^{2}):=\text{The space of real entire functions on $\mathbb R^{2}$}$$

Note. We note that, $RE(\mathbb R^{2}) \subset RA(\mathbb R^{2}).$

Example. There exists $$f(s,t) = \frac{1}{(1+s^{2}) (1+t^{2})}, (s,t \in \mathbb R).$$

is real- analytic in the whole plane $\mathbb R^{2}$ but not real-entire; that is, $f\in RA(\mathbb R^{2})$ but $f\notin RE(\mathbb R^{2}).$

My naive questions are:

(1) How one can construct few more examples $f$ so that $f\in RA(\mathbb R^{2})$ but $f\notin RE(\mathbb R^{2})$ ?

(2) Can we think of some well-known function space say $E$, so that, $E\subset RA(\mathbb R^{2})\setminus RE(\mathbb R^{2})$ ?

(3) Can we expect to characterize the set $RA(\mathbb R^{2})\setminus RE(\mathbb R^{2})$(=The space of functions which is real analytic but not real entire) ?

Thanks,

Answer 1

1. Take a function $F$ of two complex variables $z,w$ such that $F$ is holomorphic on an open subset of $\mathbb C^2$ containing $\mathbb R^2$, but is not holomorphic on all of $\mathbb C^2$. Then $F$ is real analytic on $\mathbb R^2$ but is not real entire. Indeed, if the series $\sum_{n,m=0}^{\infty} a_{nm}(s-s_{0})^{m} (t-t_{0})^{n}$ converged absolutely for all $s,t\in \mathbb R^2$ it would also converge absolutely for all $(s,t)\in \mathbb C^2$, thus defining a function holomorphic on $\mathbb C^2$. Concrete examples can be obtained by taking any combination of polynomial and exponential functions in $z,w$, divided by a polynomial that does not vanish on $\mathbb R^2$ -- such as $(s^2+1)(t^2+1)$ in your example, or $s^4+t^6+1$, and so forth.

2. No. Not only I that can't, but I'm sure there is no well-known space of functions that requires real analyticity yet excludes real-entire functions.

3. The construction in item 1 is exhaustive: all RA\RE functions arise in this way.