Remarks on restriction eigenfunctions in \(\mathbb C^n\)
Assume that~\(\Omega\) is a bounded domain of~\(\mathbb{C}^n\) and that \(\mathcal{H}(\Omega)\) is a reproducing kernel Hilbert space of analytic functions on~\(\Omega\). Given a positive measure~\(\mu\) that is compactly supported by~\(\Omega\), the authors introduce the restriction operator \[ R: \mathcal{H}(\Omega) \rightarrow L^2(\mu) \] and investigate the operator \[ R^* R : \mathcal{H}(\Omega) \rightarrow \mathcal{H}(\Omega). \] This is a selfadjoint, positive, and compact operator with eigenvalues~\(\lambda_0 \geq \lambda_1 \geq \cdots \lambda_k \cdots\to 0\), and the main aim of the paper is to study qualitative properties of the corresponding eigenfunctions \(f_k\), which form an orthogonal system both in~\(\mathcal{H}(\Omega)\) and in \(L^2(\mu)\). The authors demonstrate that some properties observed for the model example of the Hardy space of the disk need not hold in general. The case of a measure~\(\mu\) of finite support is studied in detail. Certain extremal properties of the eigenfunctions are established. The results are illustrated by numerous examples.
scientific article; zbMATH DE number 5134498