Polynomial activation convolutions have a common degree bound

Let $X=D_{[-1,1]}$ with its fixed-support Fréchet topology. For each $n\in {\mathbb{N}}_0$, set $E_n=\{\phi \in X:g_{\phi }^{(n+1)}(x)=0\text{ for every }x\in \mathbb{R}\}$. Linearity of convolution makes $E_n$ a linear subspace. For fixed $x$, its defining functional is continuous: on the compact interval $x-[-1,1]$, local essential boundedness of $\sigma$ gives a finite bound $M_x$, and $\mathrm{\mid }{g}_{\phi }^{(n+1)}(x)-g_{\psi }^{(n+1)}(x)\mathrm{\mid }\le 2M_x{p}_{n+1}(\phi -\psi )$. Thus $E_n$, an intersection of kernels of continuous functionals, is closed.

If a smooth function $g$ has $(n+1)$st derivative zero, [F6] first makes $g^{(n)}$ constant; subtracting its corresponding multiple of $x^n\mathrm{/}n!$ leaves an $n$th derivative equal to zero. Repeating this argument down to order zero shows $g$ is a polynomial of degree at most $n$. Conversely every such polynomial has $(n+1)$st derivative zero. Thus the hypothesis gives $X={\bigcup }_{n=0}^{\mathrm{\infty }}{E}_n$.

By [F2]–[F4], the Baire theorem applies to $X$. Therefore some closed subspace $E_N$ has nonempty interior; otherwise the closed sets $E_n$ would all be nowhere dense and their countable union could not equal $X$. A linear subspace with nonempty interior contains a neighborhood of zero, and scaling any vector into that neighborhood shows it is the whole space. Hence $E_N=X$.

Membership $E_N=X$ says $g_{\phi }^{(N+1)}=0$ for every test function supported in $[-1,1]$. Repeated application of [F6] then makes every $g_{\phi }$ a polynomial of degree at most $N$, as claimed.