Determining a Function from Its Mean Values Over a Family of Spheres
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- 1 January 2004
- journal article
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Mathematical Analysis
- Vol. 35 (5) , 1213-1240
- https://doi.org/10.1137/s0036141002417814
Abstract
Suppose D is a bounded, connected, open set in Rn and f is a smooth function on Rn with support in D. We study the recovery of f from the mean values of f over spheres centered on a part or the whole boundary of D. For strictly convex D, we prove uniqueness when the centers are restricted to an open subset of the boundary. We provide an inversion algorithm (with proof) when the mean values are known for all spheres centered on the boundary of D, with radii in the interval (0, diam(D)/2). We also give an inversion formula when D is a ball in Rn, n ≥ 3 and odd, and the mean values are known for all spheres centered on the boundary.Keywords
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