# Convolution Operators on Linear Spaces of Entire Functions

Chapter
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 282)

## Abstract

Suppose that f(z) is an entire function and μ is a measure in ℂ n 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 $$\hat \mu (f)\,=\,\sum\limits_{v = 1}^s {{\lambda _v}f(z - {z^{(v)}})}$$.

## Keywords

Linear Space Entire Function Dual Space Formal Power Series Convolution Operator
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