Nikolskii-type inequalities for shift invariant function spaces

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

$\Vert f\Vert _{L_{\infty }[a + \delta ,b-\delta ]} \leq 2^{2/p^{2}} \left (\frac{n+1}{\delta } \right )^{1/p} \Vert f\Vert _{L_{p}[a,b]}$

for every $f \in V_{n}$ and $\delta \in \left (0,\frac{1}{2}(b-a) \right )\,.$