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An inversion formula for the spherical transform in $$S^{2}$$ S 2 for a special family of circles of integration

An inversion formula for the spherical transform in $$S^{2}$$ S 2 for a special family of... In this article, an inversion formula is obtained for the spherical transform which integrates functions, defined on the unit sphere $$S^{2}$$ S 2 , on circles. The inversion formula is for the case where the circles of integration are obtained by intersections of $$S^{2}$$ S 2 with hyperplanes passing through a common point $$\overline{a}$$ a ¯ strictly inside $$S^{2}$$ S 2 . In particular, this yields inversion formulas for two well-known special cases. The first inversion formula is for the special case where the family of circles of integration consists of great circles; this formula is obtained by taking $$\overline{a} = 0$$ a ¯ = 0 . The second inversion formula is for the special case where the circles of integration pass through a common point $$p$$ p on $$S^{2}$$ S 2 ; this formula is obtained by taking the limit $$\overline{a}\rightarrow p$$ a ¯ → p . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

An inversion formula for the spherical transform in $$S^{2}$$ S 2 for a special family of circles of integration

Analysis and Mathematical Physics , Volume 6 (1) – May 9, 2015

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Publisher
Springer Journals
Copyright
Copyright © 2015 by Springer Basel
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-015-0105-5
Publisher site
See Article on Publisher Site

Abstract

In this article, an inversion formula is obtained for the spherical transform which integrates functions, defined on the unit sphere $$S^{2}$$ S 2 , on circles. The inversion formula is for the case where the circles of integration are obtained by intersections of $$S^{2}$$ S 2 with hyperplanes passing through a common point $$\overline{a}$$ a ¯ strictly inside $$S^{2}$$ S 2 . In particular, this yields inversion formulas for two well-known special cases. The first inversion formula is for the special case where the family of circles of integration consists of great circles; this formula is obtained by taking $$\overline{a} = 0$$ a ¯ = 0 . The second inversion formula is for the special case where the circles of integration pass through a common point $$p$$ p on $$S^{2}$$ S 2 ; this formula is obtained by taking the limit $$\overline{a}\rightarrow p$$ a ¯ → p .

Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: May 9, 2015

References