Some inequalities for entire functions

Abstract

Let $\mathcal{L}_p$ be the subspace of the space $L_p(\mathbb{R})$ consisting of the restriction to the real axis of all entire functions of exponential type $\le \pi$. In this paper, for any function $f\in \mathcal{L}_p \,\,(1\le p\le \infty)$, we obtain estimates for the norm of $f$ in terms of the sequence $(f(n/2))_{n\in\mathbb{Z}},$ namely
$$
\frac12 \|f\|_{p,1}\le \|f\|_{\mathcal{L}_p}\le 2 \|f\|_{p,1},
$$
where $\|f\|_{p,1}:=\frac12(\|Jf\|_{\ell_p(\mathbb{Z})} +\|JT_{1/2}f\|_{\ell_p(\mathbb{Z})})$. Here $J:\mathcal{L}_p\to\ell_p(\mathbb{Z})$ is the linear operator given by the formula $$ (Jf)(n):=(-1)^nf(n), \quad n\in\mathbb{Z},
$$ and $T_\tau$ is the shift by $\tau\in\mathbb{R}$ of the function $f$,
$$ (T_\tau f)(z):=f(z+\tau), \quad z\in\mathbb{C}.
$$