Convolution Operators on Linear Spaces of Entire Functions

Abstract

Suppose that f(z) is an entire function and μ is a measure in ℂⁿ with compact support. We define the convolution operator $\hat \mu$(f) by

$$\hat \mu (f)\, = \,g\,= f \star \mu \, = \,\mathop \smallint \limits_{{\mathbb{C}_n}} \,f(z + w)d\mu(w)$$

((9,1))

It is a simple consequence of Cauchy’s Theorem in the polydisc, for instance, that this includes all finite order differential operators with constant coefficients, and if we choose $\mu \, = \,\sum\limits_{v = 1}^s {{\lambda _v}\delta ({z^{(v)}})}$, δ the Dirac measure, then we obtain the finite difference operator $\hat \mu (f)\,=\,\sum\limits_{v = 1}^s {{\lambda _v}f(z - {z^{(v)}})}$.