Does the Fourier transform preserve the separation property?

Question

The space of Schwartz functions on the plane is denoted by $\mathcal{S}$. The usual multiplication and the convolution multiplication on $\mathcal{S}$ are denoted by $m_1$ and $m_2$, respectively.

The Fourier transform $\mathcal{F}$ on $\mathcal{S}$ give a bijective correspondence between the $m_1$-subalgebras of $\mathcal{S}$ and $m_2$- subalgebras of $\mathcal{S}$.

We say that a subset $A$ of $\mathcal{S}$ separates compact subsets of $\mathbb{R}^2$ if for every disjoint compact set $K_1,K_2, \ldots,K_n$, there exist a function $f\in A$ such that $f(K_i) \cap f(K_j)$ is null, for $i\neq j$.

Is it true to say that the Fourier transform gives a bijective correspondence between separating $m_1$-subalgebras of $\mathcal{S}$ and its $m_2$ separating subalgebras?

The question is motivated by the following post:

Are these function spaces appropriate to be considered as the domain of certain differential operator?

Answer 1

I can't give a precise or complete answer, but based on vague, spontaneous thoughts I suggest considering the subalgebra of the Schwartz space generated by all bump functions of the form $$ f^c_x(y):=e^{-\frac{c}{\Vert x-y\Vert^2}},\quad x,y\in \mathbb{R}^2,\; c>0. $$ Clearly, this algebra is separating. In these notes it is shown that the Fourier transform $\hat f^1_x$ of $f^1_x$ fulfills $$ \hat f^1_x(\xi)\sim 2 \mathrm{Re}\,\bigg(\sqrt{\frac{-i\pi}{\sqrt{2i}\xi^{3/2}}}e^{i\xi-1/2-\sqrt{2i\xi}}\bigg)\qquad \text{as }\xi \to +\infty. $$ Deducing from this the asymptotic behaviour of the Fourier transforms of the functions $f^c_x$ for $c\neq 1$, one should be able to conclude that the Fourier transforms $\hat f^c_x(\xi)$ oscillate very quickly as $\xi$ becomes large. This should imply that also their convolutions oscillate very quickly as $\xi$ becomes large, hence the algebra formed by the $\hat f^c_x(\xi)$ under convolution is not separating anymore because when one chooses two compact sets that are large enough and far enough away from the origin, then there won't be any two functions in the algebra that don't oscillate at least once inside each of the two compact sets.

Of course, when looking more closely at things I might be totally wrong and these considerations might be misleading, but maybe they can at least start some discussions.