Entire functions of exponential type and model subspaces in~$H^p$

K. M. Dyakonov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 19, Tome 190 (1991), pp. 81-100

Résumé

Let $W_\sigma^p$ denote the space of all entire functions $f$ (in $\mathbb{C}$) of exponential type $\leqslant\sigma$, whose restrictions $f\mid\mathbb{R}$ belong to $L^p(\mathbb{R})$. For an inner function $\theta$ in the upper halfplane $\mathbb{C}_+$ let $K_\theta^p$ ($p\geqslant1$) be the star invariant subspace (or the model subspace) of $H^p$ generated by $\theta$: $K_\theta^p\stackrel{def}=H^p\cap\theta\overline{H^p}$, where $H^p=H^p(\mathbb{C}_+)$ is the usual Hardy class. It is shown that many well-known properties of the spaces $W_\sigma^p$ (e.g. some imbedding and uniqueness theorems, the Logvinenko–Sereda theorem about equivalent norms, the S. N. Bernstein differential inequality) hold for $K_\theta^p$ if and only if the derivative $\theta'$ is bounded. The classical results on entire functions are obtained by setting $\theta(x)=\exp(i\sigma x)$.