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
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 \(\hat \mu (f)\,=\,\sum\limits_{v = 1}^s {{\lambda _v}f(z - {z^{(v)}})}\).
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© 1986 Springer-Verlag Berlin Heidelberg
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Lelong, P., Gruman, L. (1986). Convolution Operators on Linear Spaces of Entire Functions. In: Entire Functions of Several Complex Variables. Grundlehren der mathematischen Wissenschaften, vol 282. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-70344-7_9
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DOI: https://doi.org/10.1007/978-3-642-70344-7_9
Publisher Name: Springer, Berlin, Heidelberg
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