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.
Dedicated to Wolfgang Sprößig on the occasion of his 70th birthday
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References
F. Brackx, B. De Knock, H. De Schepper, D. Eelbode, A calculus scheme for Clifford distributions. Tokyo J. Math. 29(2), 495–513 (2006)
F. Brackx, F. Sommen, J. Vindas, On the radial derivative of the delta distribution. Complex Anal. Oper. Theory 11(5), 1035–1057 (2017)
R. Delanghe, F. Sommen, V. Souček, Clifford Algebra and Spinor–Valued Functions: A Function Theory for the Dirac Operator (Kluwer Academic Publishers, Dordrecht, 1992)
K. Gürlebeck, K. Habetha, W. Sprössig, Holomorphic Functions in the Plane and n-Dimensional Space, translated from the 2006 German original (Birkhäuser Verlag, Basel, 2008)
S. Helgason, The Radon Transform (Birkhäuser, Boston, 1999)
I. Porteous, Clifford Algebras and the Classical Groups (Cambridge University Press, Cambridge, 1995)
L. Schwartz, Théorie des Distributions (Hermann, Paris, 1966)
E.M. Stein, G. Weiss, Generalization of the Cauchy-Riemann equations and representations of the rotation group. Am. J. Math. 90, 163–196 (1968)
Đ. Vučković, J. Vindas, Rotation invariant ultradistributions, in Generalized Functions and Fourier Analyis, Operator Theory: Advances and Applications (Springer, Basel, 2017)
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Brackx, F. (2019). Radial and Angular Derivatives of Distributions. In: Bernstein, S. (eds) Topics in Clifford Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-23854-4_7
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DOI: https://doi.org/10.1007/978-3-030-23854-4_7
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