A representation theorem for functions holomorphic off the real axis

Abstract:

Let f be holomorphic in the union of the upper and lower half planes, and let $p \in [1,\infty )$. We prove that there exists an entire function $\varphi$ and a sequence $\{ {f_n}\}$ in ${L^p}(R)$ satisfying $\left \| {{f_n}} \right \|_p^{1/n} \to 0$ such that \[ f(z) = \varphi (z) + \sum \limits _{n = 0}^\infty {\int _{ - \infty }^\infty {{{(t - z)}^{ - n - 1}}{f_n}(t)dt.} } \] This complements an earlier result of the author’s on representation of function holomorphic outside a compact subset of the Riemann sphere. A principal tool in both proofs is the Köthe duality between the spaces of functions holomorphic on and off a subset of the sphere. A corollary of the present result is that each hyperfunction of one variable can be represented by a sum of Cauchy integrals over the real axis.