### 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 $z_1, \ldots, z_n \in D$, and $w_1, \ldots, w_n \in \mathbb{C}$ be given. There exists a function $f \in H^\infty(D)$ satisfying $\|f\|_\infty \leq 1$ and

$f(z_i) = w_i \,\, \,\, i=1, \ldots, n$

if and only if the following matrix inequality holds:

$\big(\frac{1-w_i \overline{w_j}}{1 - z_i \overline{z_j}} \big)_{i,j=1}^n \geq 0 .$

Note that the matrix element $\frac{1-w_i\overline{w_j}}{1-z_i\overline{z_j}}$ appearing in the theorem is equal to $(1-w_i \overline{w_j})k(z_i,z_j)$, where $k(z,w) = \frac{1}{1-z \overline{w}}$ is the reproducing kernel for the Hardy space $H^2$ (this kernel is called the Szego kernel). Given $z_1, \ldots, z_n, w_1, \ldots, w_n$, the matrix

$\big((1-w_i \overline{w_j})k(z_i,z_j)\big)_{i,j=1}^n$

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).