Nevanlinna–Pick interpolation

Remarks

Proof idea for necessity: if $\abs{f} \le 1$, the multiplication operator by $f$ on the Hardy space $H^2$ has norm at most $1$. Its adjoint sends the reproducing kernel $k_{z_i}(z) = \frac{1}{1 - \overline{z_i} z}$ to $\overline{w_i} k_{z_i}$, and the condition $\norm{M_f^* g} \le \norm{g}$ on combinations of these kernels is exactly $P \ge 0$. Sufficiency can be proved by Pick's original induction, which reduces $n$ points to $n - 1$ points with a Möbius change of variables, or by the commutant lifting theorem.

Example: for $n = 2$ with $z_1 = w_1 = 0$, $z_2 = a \ne 0$ and $w_2 = b$, the matrix is positive semidefinite exactly when $\abs{b} \le \abs{a}$. This is the Schwarz lemma; when $\abs{b} = \abs{a}$, the matrix has rank $1$ and the unique solution is $f(z) = (b/a) z$.

Going deeper

For a student

The theorem generalises the Schwarz–Pick lemma from one pair of points to any finite set. Its operator form says that multipliers of $H^2$ of norm at most $1$ are characterised by positivity of a kernel. This viewpoint later led to the theory of complete Pick spaces. The same method treats interpolation in the upper half-plane and boundary interpolation problems, and it has applications in control theory. For infinitely many points the condition becomes positivity of an infinite Pick matrix, and the interpolating sequences are characterised by Carleson.