The Pick Problem and Related Function Spaces

Series: 
Research Horizons Seminar
Wednesday, March 5, 2014 - 12:00
1 hour (actually 50 minutes)
Location: 
Skiles 005
,  
School of Math
The classic Pick Interpolation Problem asks: Given points z_1, z_n and w_1, w_n in the unit disk, is there a function f(z) that (1) is holomorphic on the unit disk, (2) satisfies f(z_i)=w_i, and (3) satisfies |f(z)|=1 In 1917, Pick showed that such a function f(z) exists precisely when an associated matrix is positive semidefinite. In this talk, I will translate the Pick problem to the language of Hilbert function spaces and present a more modern proof of the Pick problem. The benefit of this approach is that, as shown by J. Agler in 1989, it generalizes easily to the two-variable setting. At the heart of the proof is a method of representing bounded analytic one and two-variable functions using Hilbert space operators. Time-permitting, I will discuss recent results concerning the structure of such representations for bounded two-variable analytic functions, which is joint work with G. Knese.