Pick interpolation is a classical subject that ties together function theory and operator algebras. A constrained Pick interpolation problem is given by a set of points and values in the unit disc, along with a set of constraints. Davidson, Paulsen, Raghupathi, and Singh have solved the constrained Nevanlinna-Pick problem in the case of a single constraint $f'(0) = 0$. They have introduced a family of kernels parametrized by a sphere, such that the positivity of the corresponding Pick matrices is equivalent to the existence of a solution. Ragupathi, in his thesis, and Dritschel and his collaborators have solved a wide variety of cases.

I will introduce a more algebro-geometric approach to this problem and show how the results of Davidson and Hamilton combined with the above techniques yield a family of kernels for any contained Pick interpolation problem with finitely many constraints.

This talk is based on joint work with Ken Davidson.