Minimal $H^{2}$ interpolation in the Carathéodory class

Abstract:

For $({c_1}, \ldots ,{c_n})$ in ${{\mathbf {C}}^n}$, let $C({c_1}, \ldots ,{c_n})$ denote the class of functions $f(z) = 1 + {c_1}z + \cdots + {c_n}{z^n} + \Sigma _{k = n + 1}^\infty {a_k}{z^k}$ which are analytic and satisfy $\operatorname {Re} f(z) > 0$ in the unit disc. The unique function of least ${H^2}$ norm in $C({c_1}, \ldots ,{c_n})$ is explicitly determined.