On sets that are uniquely determined by a restricted set of integrals

Author:
J. H. B. Kemperman

Journal:
Trans. Amer. Math. Soc. **322** (1990), 417-458

MSC:
Primary 44A60; Secondary 28A99, 49R99, 60A10

DOI:
https://doi.org/10.1090/S0002-9947-1990-1076178-4

MathSciNet review:
1076178

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Abstract: In many applied areas, such as tomography and crystallography, one is confronted by an unknown subset of a measure space such as , or an unknown function on , having known moments (integrals) relative to a specified class of functions . Usually, these -moments do not fully determine the object or function . We will say that is a set of uniqueness if no other function has the same -moments as in so far as the latter moments exist. Here, is identified with its indicator function.

Within this general setting and with no further assumptions, we develop a powerful sufficient condition for uniqueness, called generalized additivity. When is a finite class, this condition of generalized additivity is shown to be also necessary for uniqueness. For each , which is not the indicator function of a set of uniqueness, there exist infinitely many sets having the same -moments as , provided is nonatomic or regular and, moreover, `strongly rich', a condition which is satisfied in many applications.

Using such general results, we also study the uniqueness problem for measures with given marginals relative to a given system of projections . Here, one likes to know, for instance, what subsets of are uniquely determined by the corresponding set of projections (*X*-ray pictures). It is allowed that . Our results are also relevant to a wide class of optimization problems.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1990-1076178-4

Keywords:
Sets with given moments,
generalized additive sets,
sets of uniqueness,
reconstructing a set from its projections,
strongly rich systems,
optimal bang-bang control,
tomography

Article copyright:
© Copyright 1990
American Mathematical Society