Nikolskii-type inequalities for shift invariant function spaces

Abstract:

Let $V_{n}$ be a vectorspace of complex-valued functions defined on ${\mathbb {R}}$ of dimension $n+1$ over ${\mathbb {C}}$. We say that $V_{n}$ is shift invariant (on ${\mathbb {R}}$) if $f \in V_{n}$ implies that $f_{a} \in V_{n}$ for every $a \in {\mathbb {R}}$, where $f_{a}(x) := f(x-a)$ on ${\mathbb {R}}$. In this note we prove the following.

Theorem. Let $V_{n} \subset C[a,b]$ be a shift invariant vectorspace of complex-valued functions defined on ${\mathbb {R}}$ of dimension $n+1$ over ${\mathbb {C}}$. Let $p \in (0,2]$. Then \begin{equation*}\|f\|_{L_{\infty }[a + \delta ,b-\delta ]} \leq 2^{2/p^{2}} \left (\frac {n+1}{\delta } \right )^{1/p} \|f\|_{L_{p}[a,b]} \end{equation*} for every $f \in V_{n}$ and $\delta \in \left (0,\frac {1}{2}(b-a) \right ) .$