Вронскианы и глубокие нули голоморфных функций

Let $X$ be a finite-dimensional subspace of $H(G)$, the space of holomorphic functions on a planar domain $G$. Then there is a discrete subset $S=S(X)$ of $G$ that contains every ‘deep’ zero of every nontrivial function in $X$. (Here, ‘deep’ should be understood as having multiplicity greater than or equal to the dimension of $X$). We elaborate on this by studying similar phenomena, but with more sophisticated boundary smallness conditions playing the role of deep zeros.