On sets that are uniquely determined by a restricted set of integrals
Author:
J. H. B. Kemperman
Journal:
Trans. Amer. Math. Soc. 322 (1990), 417458
MSC:
Primary 44A60; Secondary 28A99, 49R99, 60A10
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 (Xray pictures). It is allowed that . Our results are also relevant to a wide class of optimization problems.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199010761784
PII:
S 00029947(1990)10761784
Keywords:
Sets with given moments,
generalized additive sets,
sets of uniqueness,
reconstructing a set from its projections,
strongly rich systems,
optimal bangbang control,
tomography
Article copyright:
© Copyright 1990
American Mathematical Society
