On the Carathéodory–Schur interpolation problem over quaternions

We consider the quaternion version of the Toeplitz matrix extension problem with prescribed number of negative eigenvalues. The positive semidefinite case is closely related to the Carathéodory–Schur interpolation problem (CSP) in the Schur class \(\mathcal S_{{\mathbb {H}}}\) and the Carathéodory class \({\mathcal {C}}_{{\mathbb {H}}}\) of slice-regular functions on the unit quaternionic ball \({\mathbb {B}}\) that are, respectively, bounded by one in modulus and having positive real part in \({\mathbb {B}}\). Explicit linear fractional parametrization formulas with free Schur-class parameter for the solution set of the CSP (in the indeterminate case) are given. Carathéodory–Fejér extremal problem and Carathéodory theorem on uniform approximation of a Schur-class function by quaternion finite Blaschke products are also derived. The indefinite version of the Toeplitz extension problem is applied to solve the CSP in the quaternion generalized Schur class. The linear fractional parametrization of the solution set for the indefinite indeterminate problem still exists, but some parameters should be excluded. These excluded parameters and the corresponding “quasi-solutions" are classified and discussed in detail.