Wronskians and deep zeros of holomorphic functions (Q387977)

Let \(\Omega\) be a domain in the complex plane \(\mathbb{C}\) and \(\mathcal{H}(\Omega)\) denote the set of holomorphic functions on \(\Omega\). For a non-null function \(f\in \mathcal{H}(\Omega)\) it is well known that its zero set \(\mathcal{Z}(f)=\{z\in \Omega :\, f(z)=0\}\) is discrete in \(\Omega\). The author defines a point \(z_0\in \mathcal{Z}(f)\) as ``\(n\)-deep for \(f\)'' if its multiplicity is at least \(n+1\) and shows that if \(f_0,\dots,f_n \in \mathcal{H}(\Omega)\) are linearly independent on \(\Omega\) and \(\lambda_0,\dots ,\lambda_n\) are complex numbers not all equal to zero, then there is a discrete subset \(\mathcal{E}\) of \(\Omega \) such that the \(n\)-deep zeros of the function \(\sum_{j=0}^n \lambda _jf_j\) are all contained in \(\mathcal{E}\).NEWLINENEWLINEThe author extends his very interesting study to various function spaces on the unit disk for which he introduces more sophisticated boundary smallness conditions playing the role of deep zeros.