Integral Representation of Holomorphic Functions on Banach Spaces

Ignacio Zalduendo

Abstract

The Cauchy integral formula has no true analogue in infinite-dimensional holomorphy. The usual generalisation, though quite useful, is essentially the one-dimensional formula in each direction: for each x, and |z| < r,

f(zx) = \frac{1}{2πi} ∫_{|λ|=r} \frac{f(λx)}{λ − z}dλ.

However, integral expressions valid for some homogeneous polynomials and some holomorphic functions have been proposed. All of them involve integration over E′ rather than E. An integral k-homogeneous polynomial over E is

P (x) = ∫_{B_{E′}} γ(x)^{k}dμ(γ),

and an integral holomorphic function f : B_{E}^{∘} ⟶ C is

f(x) = ∫_{B_{E′}} \frac{1}{1 − γ(x)}dμ(γ).

In these expressions, the measure μ is said to represent the polynomial P or the function f, but there are many such representing μ’s, and little has been said in the way of expressing these measures in terms of the functions and a universal measure G, i.e.: dμ(γ) = f̃(γ)dG(γ).

In this talk we discuss this problem for a class of entire functions of exponential type over a Banach space E with separable dual. We obtain integral representations of the form

f(x) = ∫_{E′} e^{γ(x)}f̃(γ)dG(γ)

where G is a Gaussian measure on E′ and f̃is a transformation of f involving the covariance operator of G.