Integral representation of holomorphic functions on Banach spaces (Q557863)

In this nice paper, the authors present an integral representation of certain classes of holomorphic functions \(f\) on Banach spaces \(E\) in two alternative forms: \[ f(x)=\int_{E'} \exp(\langle x,x'\rangle)\cdot \overline{f_1(x')}~dW(x') \] and \[ f(x)=\int_{E'} \frac{1}{1-\langle x,\frac{x'}{\| x'\| }\rangle}\overline{f_2(x')}~dW(x'), \] which could be considered as the Banach space version of the Cauchy integral formula. Here, \(W\) is an abstract Wiener measure on \(E'\) and \(f_1,f_2\) are certain transformations of \(f\) involving the covariance operator \(A:E''\to E'\) of the measure \(W\). An important class of holomorphic functions to which the integral representation formula applies is the class of all entire holomorphic functions \(f:E\to \mathbb{C}\) of \(A\)-exponential type, i.e., \(| f(x)| \leq c\cdot \exp(\sigma\| Ax\| )\) for certain constants \(c, \sigma\) and all \(x \in E\).