Geodesic
Zero sets for functions from $\Lambda_\omega$
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part X, Tome 107 (1982), pp. 178-188 Cet article a éte moissonné depuis la source Math-Net.Ru

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The following result is proved: THEOREM: {\it Let $S$ be an inner function, $\operatorname{spec}S\subset E$, $E\subset\operatorname{clos}\mathbb D$. Suppose $E$ satisfies $$ \sum_{\alpha\in\mathbb D\cap E}(1-|\alpha|)<\infty,\quad\int_{\partial\mathbb D}\log\omega(\operatorname{dist}(z,E))|dz|>-\infty, $$ $\omega$ being a continuity modulus. Then there exists a function $\Lambda_\omega$ such that $f^{-1}(0)\in E$ и $f|_S\in\Lambda_\omega$}. 
@article{ZNSL_1982_107_a11,
     author = {N. A. Shirokov},
     title = {Zero sets for functions from $\Lambda_\omega$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {178--188},
     year = {1982},
     volume = {107},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1982_107_a11/}
}
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N. A. Shirokov. Zero sets for functions from $\Lambda_\omega$. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part X, Tome 107 (1982), pp. 178-188. http://geodesic.mathdoc.fr/item/ZNSL_1982_107_a11/