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Convolution Operators on Linear Spaces of Entire Functions

  • Pierre Lelong
  • Lawrence Gruman
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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Pierre Lelong
    • 1
  • Lawrence Gruman
    • 2
  1. 1.Université Paris VIParis Cedex 05France
  2. 2.UER de mathématiquesC.N.R.S., Université de ProvenceMarseilleFrance

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