Let $f: \mathbb R \rightarrow \mathbb R$ be a periodic, say with the period 1, and locally integrable function.
Assume that $g: \mathbb R \rightarrow \mathbb R$ is a weak derivative of $f$ of order $k$, that is $g$ is also locally integrable and satisfies condition $$ \int f D^k \phi dx=(-1)^k\int g \phi dx $$ for all smooth functions $\phi : \mathbb R \rightarrow \mathbb R$ with compact supports.
My question is if we can replace in this condition "for all smooth functions $\phi : \mathbb R \rightarrow \mathbb R$ with compact supports" by
for all smooth functions $\phi : \mathbb R \rightarrow \mathbb R$ with compact supports contained in $(0,1)$
or by
for all smooth $1$-periodic functions $\phi : \mathbb R \rightarrow \mathbb R$.
