I am reading in my lecture notes about Tempered distributions and one of the statements goes as follows:
Let $T$ be a Tempered distributions, define the shifted distribution $T_{\{a, L\}}(f)=|\operatorname{det}(L)|^{-1} T(\{a, L\} f)$, where $(\{a, L\} f)(x)=f\left(L^{-1}(x-a)\right)$, and if $T$ admits a functional representation: $T(f)=\int d x T(x) f(x)$, then $T_{\{a, L\}}(x)=T(L x+a)$.
and I am not sure how to go about proving this. any help or pointers would be appreciated. Thanks
Assuming that $T$ has a functional representation let's compute $T_{ \{ a,L\} }$.
$$T_{ \{ a,L\} }(f)=|det(L)|^{-1}T(( a,L)f)=\int |det(L)|^{-1}T(x)f(L^{-1}(x-a))dx$$ Now performing a change of variables letting $x'=L^{-1}(x-a)$ and recalling what happens to a measure when we apply an invertible linear transformation we should get the solution.