Let $M(n)$ be the vector space of $n\times n$ matrices over a local non-archimedian field $K$.
Let $\mathcal S$ denote the space of locally constant compactly supported functions
on $M(n)$. Similarly, let $\mathcal S ^0$ denote the space of locally constant compactly supported functions
on the group $GL(n)$ (which is an open subset of $M(n)$). 


Let us identify $M(n)$ with its dual space by means of the pairing $(x,y)=Tr(xy)$.
Let $F$ denote the corresponding Fourier transform acting from $\mathcal S$ to itself
(formally, $F$ depends on a choice of a non-trivial additive character of $K$, but it does not matter for the question).
Is this true that 
$$
\mathcal S=\mathcal S^0 + F(\mathcal S^0)?
$$
This is obvious when $n=1$, but already in the case $n=2$ the asnwer is not clear to me.



A question on non-archimedian Fourier transform
Actually, the answer is indeed negative and the explanation is very simple. Namely, for $n=2$ let $\phi$ be the delta-function of the space of matrices whose second row is zero (considered as a distribution). Then its Fourier transform is the delta-function of matrices whose first column is zero. Hence this is a distribution with support on degenerate matrices whose Fourier transform is also concentrated on degenerate matrices.


