Constructive description of finitely generated ideals in an algebra of functions analytic in an open disc and smooth in its closure
Denoting by \(\Lambda^a_\omega\) the class of holomorphic in the open unit disc \(D\) functions \(f\), for which \[ \bigl |f (\xi_1) - f (\xi_2) \bigr |\leq C_f \omega \bigl( |\xi_1 - \xi_2 |\bigr), \quad \xi_1, \xi_2 \in \overline D, \] where \(\omega\) is a nonnegative, increasing function satisfying Zygmund condition. The author shows that if \(g\), \(f_1, \ldots, f_n\) \[ \Lambda^a_\omega \text{ and } \bigl |g (\xi) \bigr |\leq C \biggl( \bigl |f_1 (\xi) \bigr |^2 + \cdots + \bigl |f_n (\xi) \bigr |^2 \biggr),\;\xi \in D, \] then there exist functions \(h_1, \ldots, h_n \in \Lambda^a_\omega\) such that \(g^2 (\xi) = f_1 (\xi) h_1 (\xi) + \cdots + f_n (\xi) h_n (\xi)\), \(\xi \in D\). He also shows that the converse is true in a sense. Moreover he considers the case of algebra \(\Lambda^{(m)}_\alpha\), \(0 < \alpha < 1\), \(m \geq 1\).
scientific article; zbMATH DE number 795474