Constrained Pick interpolation for multiply connected domains

The Pick interpolation theorem states that the existence of a function on the complex unit disc that is analytic, bounded by 1, and satisfies some interpolation data is equivalent to the positivity of a matrix that depends on the interpolation data. In 1979 Abrahamse generalized this result from the disk to any g-holed multiply connected domain. However, in the result of Aabrahamse, a family of matrices parametrized by the g-dimensional torus was needed. In 2010, A variation of the Pick interpolation problem was studied by Davidson, Paulsen, Raghupathi, and Singh, who discovered that if the constraint of zero derivative at a point is applied to the interpolating function, then there is a family of matrices parametrized by the unit sphere that need to be positive. In my thesis, I have combined these results to solve a constrained interpolation problem on a multiply connected domain. I will present the ideas that prove these kinds of interpolation theorems, that were first applied to that cause by Sarason, and will show how I used them for the constraint-multiply connected problem. If time allows it, I will also say a few words about matrix-valued interpolation.