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 Ω \Omega of K and vanishing at infinity. We prove that there are finite complex-valued Borel measures μ m , n ( m , n = 0 , 1 , 2 , … ; m + n ⩾ 1 ) {\mu _{m,n}}(m,n = 0,1,2, \ldots ;m + n \geqslant 1) on K 2 {K^2} satisfying lim k → ∞ ( Σ m + n = k ‖ μ m , n ‖ ) 1 / k = 0 {\lim _{k \to \infty }}{({\Sigma _{m + n = k}}\left \|{\mu _{m,n}}\right \|)^{1/k}} = 0 so that \[ f ( z ) = ∑ m , n ∫ K 2 ( z − w 1 ) − m ( z − w 2 ) − n d μ m , n ( w 1 , w 2 ) ( z ∈ Ω ) . f(z) = \sum \limits _{m,n} {\int _{{K^2}} {{{(z - {w_1})}^{ - m}}{{(z - {w_2})}^{ - n}}d{\mu _{m,n}}({w_1},{w_2})\quad (z \in \Omega ).} } \]