Jet functions having indefinite Carathéodory–Pick matrices Original Research Article

A class of functions is introduced that take values in the set of ordered tuples of complex numbers and are defined on a subset of the unit disc; the number of components of the value of a function at a given point may be countably infinite and may depend on the point. The class is defined by the property that all Carathéodory–Pick matrices of a function have not more than a prescribed number of negative eigenvalues, and at least one Carathéodory–Pick matrix of the function has exactly the prescribed number of negative eigenvalues. The class is characterized in several ways. It turns out that a typical function in the class is generated by a meromorphic function, together with several of its derivatives at regular points, with a possible modification at a finite number of points. Extension and interpolation results are proved for functions in the class. These functions are also interpreted as pseudomultipliers on the the Hardy space H2 of the unit disc.