Jet functions having indefinite Carathéodory-Pick matrices
Given a nonnegative integer \(\kappa\) and a nonempty subset \(\Omega\) of the open unit disk, the class \(\mathcal{P}_\kappa(\Omega)\) consists of complex valued functions \(f\) with the domain of definition \(\Omega\) such that all Pick matrices \[ P(f;z_1,\ldots,z_n):= \left[ \frac{1-f(z_i)f^*(z_j)}{1-z_iz^*_j}\right]_{i,j=1}^n, \qquad z_1,\ldots,z_n\in\Omega \] have at most \(\kappa\) negative eigenvalues and at least one such matrix has exactly \(\kappa\) negative eigenvalues. It was a matter of preceding papers by the authors to characterize the functions from \(\mathcal{P}_\kappa(\Omega)\) in term of their ``meromorphic'' extension. In the present paper a class of functions \(f\) called jet valued functions such that \(f\) belongs to \(\mathbb{C}^{m(z)}\), where \(0\leq m\leq\infty\) is an integer valued function, and all Carathéodory-Pick matrices enjoy the same property as the aforementioned one, is introduced and characterized in several ways. Extension and interpolation results are proved for functions in the class.
scientific article; zbMATH DE number 2083516
scientific article