Convolution Operators on Linear Spaces of Entire Functions

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


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
$$\hat \mu (f)\, = \,g\,= f \star \mu \, = \,\mathop \smallint \limits_{{\mathbb{C}_n}} \,f(z + w)d\mu(w)$$
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 \(\hat \mu (f)\,=\,\sum\limits_{v = 1}^s {{\lambda _v}f(z - {z^{(v)}})}\).


Linear Space Entire Function Dual Space Formal Power Series Convolution Operator 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  1. 1.Université Paris VIParis Cedex 05France
  2. 2.UER de mathématiquesC.N.R.S., Université de ProvenceMarseilleFrance

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