|
Zapiski Nauchnykh Seminarov LOMI, 1982, Volume 107, Pages 27–35
(Mi znsl3413)
|
|
|
|
This article is cited in 4 scientific papers (total in 4 papers)
On Taylor coefficients and $L_p$-continuity moduli of Blaschke products
I. É. Verbitskii
Abstract:
Let
$$
B=\prod_{k\geq1}b_{z_k},\quad b_{z_k}\overset{\text{def}}=\frac{|z_k|}{z_k}
\frac{z_k-z}{1-\bar z_k z},\quad |z_k|<1,
$$
be a Blaschke product, let $\widehat{B}(z)$ denote its $k$-th Taylor
coefficient. Suppose $\{z_k\}$ splits into a finite union of sequences
$\{\xi_k\}$ satisfying
$$
\sup_{k\geq1}\frac{1-|\xi_{k+1}|}{1-|\xi_{k}|}<1.
$$
The following assertions are proved to be equivalent:
1. $\{z_k\}_{k\geq1}\in(\omega N)$;
2. $\widehat{B}(k)=O(1/k)$, $k\to\infty$;
3. $\sum_{k\geq n}|\widehat{B}|^2=O(n^{-1})$;
4. $B\in\operatorname{Lip}(1/p,L^p)$ for some $p$, $1<p<\infty$.
Citation:
I. É. Verbitskii, “On Taylor coefficients and $L_p$-continuity moduli of Blaschke products”, Investigations on linear operators and function theory. Part X, Zap. Nauchn. Sem. LOMI, 107, "Nauka", Leningrad. Otdel., Leningrad, 1982, 27–35; J. Soviet Math., 36:3 (1987), 314–319
Linking options:
https://www.mathnet.ru/eng/znsl3413 https://www.mathnet.ru/eng/znsl/v107/p27
|
Statistics & downloads: |
Abstract page: | 453 | Full-text PDF : | 103 |
|