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Radial and Angular Derivatives of Distributions

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Part of the Trends in Mathematics book series (TM)

Abstract

When expressing a distribution in Euclidean space in spherical co-ordinates, derivation with respect to the radial and angular co-ordinates is far from trivial. Exploring the possibilities of defining a radial derivative of the delta distribution \(\delta ( \underline {x})\) (the angular derivatives of \(\delta ( \underline {x})\) being zero since the delta distribution is itself radial) led to the introduction of a new kind of distributions, the so-called signumdistributions, as continuous linear functionals on a space of test functions showing a singularity at the origin. In this paper we search for a definition of the radial and angular derivatives of a general standard distribution and again, as expected, we are inevitably led to consider signumdistributions. Although these signumdistributions provide an adequate framework for the actions on distributions aimed at, it turns out that the derivation with respect to the radial distance of a general (signum)distribution is still not yet unambiguous.

Keywords

Distribution Radial derivative Angular derivative Signumdistribution 

Mathematics Subject Classification (2010)

Primary 46F05 46F10; Secondary 30G35 

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematical AnalysisFaculty of Engineering and Architecture, Ghent UniversityGhentBelgium

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