The best constant of the Sobolev inequality
\begin{align*}
\|\, u(x) \,\|_{\infty}\,\leq\, C \,\left\|\,u^{(M)}(x)\,\right\|_p \qquad (p\geq 1)
\end{align*}
is given by $L^q$-norm ($1/p+1/q=1$) of the well-known 
Bernoulli polynomials, which is a Green fucntion of a 
certain periodic boundary value problem for $M$-th order 
linear ordinary differential equation. 
The special case of $p=2$ is treated completely in \cite{KYWNT}.