Proof for \(p=2\) Link to heading
We have
\[ \lVert f^{(\alpha)}\rVert_2=\lVert \widehat{f^{(\alpha)}}\rVert_2 =\lVert y^\alpha \hat{f}\rVert_2 \le b^{\alpha}\lVert \hat{f}\rVert_2 = b^\alpha\lVert f\rVert_2. \]where we used Plancherel, differentiation rules, the fact that \(\supp \hat{f} \subseteq [-b,b]^d\) and again Plancherel in that order.