An integral representation of holomorphic functions

Abstract:

Let K be a compact set in the complex plane and let f be a function holomorphic on the complement $\Omega$ of K and vanishing at infinity. We prove that there are finite complex-valued Borel measures ${\mu _{m,n}}(m,n = 0,1,2, \ldots ;m + n \geqslant 1)$ on ${K^2}$ satisfying ${\lim _{k \to \infty }}{({\Sigma _{m + n = k}}\left \|{\mu _{m,n}}\right \|)^{1/k}} = 0$ so that \[ f(z) = \sum \limits _{m,n} {\int _{{K^2}} {{{(z - {w_1})}^{ - m}}{{(z - {w_2})}^{ - n}}d{\mu _{m,n}}({w_1},{w_2})\quad (z \in \Omega ).} } \]