### Advanced Analysis, Notes 17: Hilbert function spaces (Pick’s interpolation theorem)

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).