Real-analytic analogue of Schwartz functions

Question

Consider the space $\mathcal{S}'$ of functions $\mathbb{R}^n\to\mathbb C$ that are (real-)analytic and with exponential decay at infinity. This is an analogue of Schwartz space, but real-analytic rather than smooth. (Exponential decay suffices for the Fourier transform being analytic.)

Consider the commutative unital ring $\mathcal{D}$ of pseudodifferential operators $D$ of the form $$(Df)(x)=\frac1{(2\pi)^n}\int\mathrm{d}^n\xi\,\exp(\mathrm ix\cdot\xi)p_D(\xi)\,\hat f(\xi)$$ where $p_D$ is an analytic function on $\mathbb R^n$ that is slowly growing (i.e. is a tempered distribution). In particular, it includes the ring $\mathbb R[\partial_1,\dotsc,\partial_n]$ of differential operators (with constant coefficients) in the special case where $p_D$ is a polynomial.

Now, I’d like the following to hold:

1. $\mathcal{S}'$ is closed under pointwise products. (That is, it forms a non-unital associative commutative algebra over $\mathbb R$.)
2. $\mathcal{S}'$ is closed under the action of $\mathcal{D}$. (That is, it forms a module over $\mathcal{D}$.)
3. An analytic Leibniz rule holds in the sense that, for $f,g\in\mathcal{S}'$ and $D=\sum_{I\in\mathbb N^n}c_I\partial_I$, $$D(fg)=\sum_{I\in\mathbb N^n}\sum_{\substack{I',I''\in\mathbb N^n\\I'+I''=I}}c_I\binom I{I'}(\partial_{I'}f)(\partial_{I''}g),$$ where $I$ is a multi-index and where convergence is taken in some suitable topology.
4. We have $\mathcal{S}'\otimes_{\mathcal{D}}\mathcal S'=\mathcal{S}'$. (That is, the evident inclusion $\mathcal{S}'\otimes_{\mathcal{D}}\mathcal S'\subset\mathcal{S}'$ is surjective.)

I'd like to know:

1. Do the above desired properties hold? If not, is there some variation of the definitions of $\mathcal{D}$ and $\mathcal{S}'$ that satisfies them?
2. Is the space $\mathcal{S}'$ or some variation thereof discussed anywhere in the literature?
3. $\mathcal{D}$ fails to be a Hopf algebra (unlike the ring of true differential operators $\mathbb R[\partial_1,\dotsc,\partial_n]\subset\mathcal{D}$) because the Leibniz rule yields infinite (rather than finite) sums. But can it be regarded as some sort of topological Hopf algebra, in which the coproduct $\Delta\colon\mathcal{D}\to\mathcal{D}\mathbin{\hat\otimes}\mathcal{D}$ takes values in some completed tensor product $\mathcal{D}\mathbin{\hat\otimes}\mathcal{D}$?

Apologies if I’ve made any mistakes. Analysis is not my usual alley.

Edit: I had originally asked the above about $\mathcal S$ being real-analytic functions whose Fourier transforms are real-analytic, but @reuns has quickly supplied the counterexample of $f=\sin(\exp(x^2))\in\mathcal S$ where $f^2\not\in\mathcal S$.

Answer 1

I'll assume that $n=1$.

I'd say that the natural choice is to consider the space $A$ of functions $f:\Bbb{R\to C}$ such that for some $c>0$, $f$ extends analytically to the strip $|\Im(z)|< c$ and $f(x+iy)=O(e^{-c|x|})$ in that strip.

Due to the Cauchy integral theorem this implies that the same holds for $\hat{f}$: for $|v|<c,|y|<c$

$$\hat{f}(u+iv)=\int_\Bbb{R} f(x)e^{-i x(u+iv)}dx = \int_\Bbb{R} f(x+iy)e^{-i (x+iy)(u+iv)}dx $$ so that $\hat{f}(u+iv)$ is holomorphic on $|v|<c$ and it is $O(e^{-uy})$.

Then it is natural to consider the space $P$ of functions $p: \Bbb{R\to C}$ that extend analytically to some strip $|\Im(z) |< d$ and such that $\forall \epsilon >0, p(x+iy)=O(e^{\epsilon |x|})$ in that strip. This way $A P=A$.

Also $AA = A$ as $u=\frac1{e^{\epsilon z}+ e^{-\epsilon z}+i}$ is in $A$ and $\forall f\in A$ taking $\epsilon $ small enough then $f/u\in A$.