×

Lacunary series and pseudocontinuations. (Russian. English summary) Zbl 0889.30029

Nikol’skij, N. K. (ed.), Studies on linear operators and function theory. 24. Work collection. Sankt-Peterburg: Matematicheskij Institut Im. V. A. Steklova, Sankt-Peterburgskoe Otdelenie RAN, Zap. Nauchn. Semin. POMI. 232, 16-32 (1996).
The principal theorem of the paper says that if \(E\subset\mathbb{Z}_+\) is a \(\Lambda(1)\)-set and \(f= \sum_{n\in E} a_nz^n\) is a holomorphic function of Nevanlinna class in the unit disc admitting a pseudocontinuation of Nevanlinna class to the annulus \(\{z\in\mathbb{C}: 1<|z|<R\}\), then \(f\) is analytic in \(\{0\leq|z|<R\}\), (so, if \(R=\infty\), \(f\) is a polynomial). The proof is based on the following result of independent interest: if \(I\) is an inner function, \(0< q<p\leq\infty\), and \(p\geq 1\), then the identical embedding \(H^p\cap I\overline H^p\to H^q\cap I\overline H^q\) is compact. Also, it is shown that if \(X\) is a subspace of \(H^1\) on which the norm convergence coincides with the convergence in measure then for \(0<q<1\) the sum \(I\overline H^q+ X\) is closed in \(L^q(\mathbb{T})\).
For the entire collection see [Zbl 0868.00022].

MSC:

30D55 \(H^p\)-classes (MSC2000)
42A55 Lacunary series of trigonometric and other functions; Riesz products