Zeros, interpolació i l'anell de funcions holomorfes en una regió

Abstract

[en] In this work, we study the construction of holomorphic functions with prescribed zeros on a domain given by the Weierstrass zeros theorem and use this result and Mittag-Leffler's theorem to interpolate a sequence of numbers by a holomorphic function. As an application of the previous topics, we study some algebraic properties of the ring $\mathcal{H}(\Omega)$ and its ideals. In particular, we prove a Bézout identity in this ring given by Wedderburn lemma. Finally, we prove Bers' theorem, which states that if the holomorphic function rings on two domains are algebraically equivalent, then the respective domains are conformally equivalent.