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| Several closed exact formulae are known for reconstruction of a function from data of spherecal means for spheres centered at a (hyper)surface. The central sets are spheres, ellipsoids or paraboloids. There is an infinite series of unknown algebraic hypersurfaces that can serve as central sets for a reconstruction formula of the same analytical form. These surfaces look exotic but have a simple algebraic definition.
It is much easier to construct a parametrix for this problem. A parametrix is a left inverse for a spherical mean operator modulo a smoothing operator in the Sobolev scale of spaces. Application of a parametrix to the integral data correctly reproduces up to a continuous function simple discontinuities like jumps or delta inclusion. |
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