Weakly Cyclic Vectors with a Given Modulus

Let H^p be the Hardy space in the polydisk. Denote by the set of all holomorphic polynomials. A vector f ∈ H^p is called weakly cyclic if the product f is weakly dense in H^p, 0 < p < 1. We construct weakly cyclic vectors with a prescribed lower semi-continuous modulus of the boundary values. Bibliography: 6 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 111–118.