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Radial and angular derivatives of distributions

Fred Brackx (UGent)
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Abstract
When expressing a distribution in Euclidean space in spherical coordinates, 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 8{x) (the angular derivatives of S(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

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MLA
Brackx, Fred. “Radial and Angular Derivatives of Distributions.” Topics in Clifford Analysis : Special Volume in Honor of Wolfgang Sprössig, edited by Swanhild Bernstein, Birkhäuser, 2019, pp. 141–71.
APA
Brackx, F. (2019). Radial and angular derivatives of distributions. In S. Bernstein (Ed.), Topics in Clifford Analysis : Special Volume in Honor of Wolfgang Sprössig (pp. 141–171). Birkhäuser.
Chicago author-date
Brackx, Fred. 2019. “Radial and Angular Derivatives of Distributions.” In Topics in Clifford Analysis : Special Volume in Honor of Wolfgang Sprössig, edited by Swanhild Bernstein, 141–71. Birkhäuser.
Chicago author-date (all authors)
Brackx, Fred. 2019. “Radial and Angular Derivatives of Distributions.” In Topics in Clifford Analysis : Special Volume in Honor of Wolfgang Sprössig, ed by. Swanhild Bernstein, 141–171. Birkhäuser.
Vancouver
1.
Brackx F. Radial and angular derivatives of distributions. In: Bernstein S, editor. Topics in Clifford Analysis : Special Volume in Honor of Wolfgang Sprössig. Birkhäuser; 2019. p. 141–71.
IEEE
[1]
F. Brackx, “Radial and angular derivatives of distributions,” in Topics in Clifford Analysis : Special Volume in Honor of Wolfgang Sprössig, 2019, pp. 141–171.
@inproceedings{8635456,
  abstract     = {When expressing a distribution in Euclidean space in spherical coordinates, 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 8{x) (the angular derivatives of S(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.},
  author       = {Brackx, Fred},
  booktitle    = {Topics in Clifford Analysis : Special Volume in Honor of Wolfgang Sprössig},
  editor       = {Bernstein, Swanhild},
  isbn         = {9783030238537},
  issn         = {2297-0215},
  keywords     = {Distribution,Radial derivative,Angular derivative,Signumdistribution},
  language     = {eng},
  pages        = {141--171},
  publisher    = {Birkhäuser},
  title        = {Radial and angular derivatives of distributions},
  url          = {http://dx.doi.org/10.1007/978-3-030-23854-4_7},
  year         = {2019},
}

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