Given a sequence of positive numbers $\{a_n\}$ and $1 < p < \infty$, $p\neq 2$, is it possible to build a function $f\in C^\infty(\mathbb R)$ so that $\|f^{(n)}\|_{L^p(\mathbb R)} = a_n$?

For what I have in mind, I would be happy with $f$ so that $\frac 1C \leq \frac{\|f^{(n)}\|_{L^p(\mathbb R)}}{a_n} \leq C$ for some $C>0$.

Thanks! Andy

Given a sequence of positive numbers $\{a_n\}$ and $1 < p < \infty$, $p\neq 2$, is it possible to build a function $f\in C^\infty(\mathbb R)$ so that $\|f^{(n)}\|_{L^p(\mathbb R)} = a_n$?

For what I have in mind, I would be happy with $f$ so that $\frac 1C \leq \frac{\|f^{(n)}\|_{L^p(\mathbb R)}}{a_n} \leq C$ for some $C>0$.

Thanks! Andy

PS I should have added that I'm really interested in sequences that grow in a very specific way, namely, for some fixed $A>0$ and $\beta>1$, $a_n = A^n n^{n\beta}$. However, I think the question is interesting regardless -- trying to know what sequences $a_n$ are allowable.

PPS I exclude $p=2$ because the function $f$ with Fourier transform $\hat f(t) = e^{-a |t|^{1/\beta}}$ works. You can compute $\|t^\ell \hat f\|_{L^2}$.