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Entire functions of exponential type and model subspaces in \(H^ p\). (Russian. English summary) Zbl 0788.30024
The author is concerned with norm inequalities in coinvariant subspaces of \(H^ p\) of the upper half-plane \((1 \leq p \leq \infty)\). For \(\theta\) an inner function for the upper half-plane, the corresponding coinvariant subspace of \(H^ p\) is denoted by \(K^ p_ \theta\) and defined to be \(H^ p \cap \theta \overline{H^ p}\). In the special case where \(\theta(z)=e^{iaz}\) \((a>0)\), the space \(K^ p_ \theta\) is contained in the space of entire functions of exponential type at most \(a\) that belong to \(L^ p\) on the real axis. The author shows that several classical inequalities pertaining to the latter spaces apply also to \(K^ p_ \theta\) provided \(\theta'\) is in \(H^ \infty\) (and only in that case). Sample result: If \(1<p \leq \infty\), ten \(\| f' \|_ p \leq \text{const} \| f \|_ p\) for all \(f\) in \(K^ p_ \theta\) if and only if \(\theta'\) is in \(H^ \infty\). This corresponds, for the case \(\theta(z)=e^{iaz}\), to a well-known inequality of S. Bernstein.

30D55 \(H^p\)-classes (MSC2000)
46E15 Banach spaces of continuous, differentiable or analytic functions
47A99 General theory of linear operators
47B38 Linear operators on function spaces (general)
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