Let $T$ be a generalized function. I need to provide definitions for $T$ to be periodic and radially symmetric.
A function (on $\mathbb{R})$is said to be periodic if there exists a $p \in \mathbb{R}$ such that $f(x+p)=f(x)$ for all $x \in \mathbb{R}$. If $T$ comes from a function then $(T_f,\phi)=\int f(x) \phi(x) dx=\int f(x+p)\phi(x) dx=\int f(x)\phi(x-p) dx=(T_f,\phi(.-p))$
So if I define $(\tau_{p}T)(\phi):=(T,\phi(.-p))$, then I think $T$ will be periodic if there exists a $p$ such that $\tau_{p}T=T$.
Similarly a function is said to be radial if $f(\theta x)=f(x)$ for $\theta$,a rotation. So i expect that if I define $(AT)(\phi)=\frac{1}{det A}(T,\phi(A^{-1}.))$, then $AT(\phi)=T(\phi)$.
Is this alright??
Thanks for the help!!