Inversion of Fractional Integrals Related to the Spherical Radon Transform
✍ Scribed by Boris Rubin
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 317 KB
- Volume
- 157
- Category
- Article
- ISSN
- 0022-1236
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✦ Synopsis
Explicit inversion formulas are obtained for the analytic family of fractional integrals (T : f )(x)=# n, : S n |xy| :&1 f ( y) dy on the unit sphere in R n+1 . Arbitrary complex : and n 2 are considered. In the ease :=0 the integral T : f coincides with the spherical Radon transform. For :>1 (:{1, 3, 5, ...) such integrals are known as the Blaschke Levy representations and arise in convex geometry, probability, and the Banach space theory. For :=1, 3, 5, ... the integral T : f is defined by continuity as the spherical convolution with the power logarithmic kernel. Different inversion methods are discussed.
1998 Academic Press
The properties (a) (c) are inherent in many other fractional integrals in one and many dimensions provided that (dÂdx) m is replaced by a suitable differential operator (see examples in [8,15]). The common feature of such integrals is the following: the dimension of the set 0 of singularities (or article no.