Activities This Week

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.

This lecture extends some standard results familiar from undergraduate calculus to the setting of higher order noncommutative functions. These extensions are accomplished using a difference-differential operator combining algebraic and topological properties. The results include the Taylor formula and an antiderivative. The Taylor formula is applied to obtain results about power series on nilpotent matrices and convergence in different topologies.