In this final lecture we will give a proof of Pick’s interpolation theorem that is based on operator theory.
Theorem 1 (Pick’s interpolation theorem): Let , and be given. There exists a function satisfying and
if and only if the following matrix inequality holds:
Note that the matrix element appearing in the theorem is equal to , where is the reproducing kernel for the Hardy space (this kernel is called the Szego kernel). Given , the matrix
is called the Pick matrix, and it plays a central role in various interpolation problems on various spaces.
I learned this material from Agler and McCarthy’s monograph [AM], so the following is my adaptation of that source.
(A very interesting article by John McCarthy on Pick’s theorem can be found here).