Does someone know where may I find the general proof of Riemann Lebesgue theorem which states that Let $1\leq p \leq \infty $ $M= \prod_{i=1}^{n} (a_i,b_i)$, and $u \in L^p$ , define $u_\nu = u(\nu x)$ and $\hat{u} = \frac{\int_M u dx}{meas M}$, then $u_n \rightarrow \hat{u}$ weakly in L^p for p>1 and converges *-weakly for $p=\infty$.

I am reading it from Darcogona's Introduction to Variational calculus, he writes that the general proof can be found in his textbook Direct methods from 1989, but I have a newer edition and I don't find this proof.

Thanks in advance.

Does someone know where may I find the general proof of Riemann Lebesgue theorem which states that Let $1\leq p \leq \infty $ $M= \prod_{i=1}^{n} (a_i,b_i)$, and $u \in L^p$ , define $u_\nu = u(\nu x)$ and $\hat{u} = \frac{\int_M u dx}{meas M}$, then $u_n \rightarrow \hat{u}$ weakly in L^p for p>1 and converges *-weakly for $p=\infty$.

I am reading it from Darcogona's Introduction to Variational calculus, he writes that the general proof can be found in his textbook Direct methods from 1989, but I have a newer edition and I don't find this proof.

Thanks in advance.

Edit: I forgot to add that there's another assumption on u and is that it may be extended by periodicity from M to all of $R^n$.