Geodesic
Weakly cyclic vectors with a~given modulus
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 31, Tome 303 (2003), pp. 111-118

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Let $H^p$ be the Hardy space in the polydisc. Denote by $\mathcal P$ the set of all holomorphic polynomials. A vector $f\in H^p$ is called weakly cyclic if the product $f\mathcal P$ is weakly dense in $H^p$, $0$. We construct weakly cyclic vectors with a prescribed lower semicontinuous modulus of the boundary values. 
@article{ZNSL_2003_303_a5,
     author = {E. Doubtsov},
     title = {Weakly cyclic vectors with a~given modulus},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {111--118},
     publisher = {mathdoc},
     volume = {303},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_303_a5/}
}
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E. Doubtsov. Weakly cyclic vectors with a~given modulus. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 31, Tome 303 (2003), pp. 111-118. http://geodesic.mathdoc.fr/item/ZNSL_2003_303_a5/