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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
MathSciNet review: 1076178
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Abstract: In many applied areas, such as tomography and crystallography, one is confronted by an unknown subset $ S$ of a measure space $ (X,\lambda )$ such as $ {{\mathbf{R}}^n}$, or an unknown function $ 0 \leqslant \phi \leqslant 1$ on $ X$, having known moments (integrals) relative to a specified class $ F$ of functions $ f:X \to {\mathbf{R}}$. Usually, these $ F$-moments do not fully determine the object $ S$ or function $ \phi $. We will say that $ S$ is a set of uniqueness if no other function $ 0 \leqslant \psi \leqslant 1$ has the same $ F$-moments as $ S$ in so far as the latter moments exist. Here, $ S$ 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 $ F$ is a finite class, this condition of generalized additivity is shown to be also necessary for uniqueness. For each $ \phi $, which is not the indicator function of a set of uniqueness, there exist infinitely many sets having the same $ F$-moments as $ \phi $, provided $ (X,\lambda ,F)$ 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 $ {\pi _j}:X \to {Y_j}(j \in J)$. Here, one likes to know, for instance, what subsets $ S$ of $ X$ are uniquely determined by the corresponding set of projections (X-ray pictures). It is allowed that $ \lambda (S) = \infty $. 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/S0002-9947-1990-1076178-4
PII: S 0002-9947(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