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.


References [Enhancements On Off] (What's this?)

  • [1] N. I. Aheizer and M. Krein, Some questions in the theory of moments, translated by W. Fleming and D. Prill. Translations of Mathematical Monographs, Vol. 2, American Mathematical Society, Providence, R.I., 1962. MR 0167806
  • [2] W. A. Carrington, Moment problems and ill-posed operator equations with convex constraints, Ph.D. Thesis, Washington Univ., St. Louis, 1982.
  • [3] P. C. Fishburn, J. C. Lagarias, J. A. Reeds, and L. A. Shepp, Sets uniquely determined by projections on axes. I. Continuous case, SIAM J. Appl. Math. 50 (1990), no. 1, 288–306. MR 1036243, 10.1137/0150017
  • [4] -, Sets uniquely determined by projections on axes II. Discrete case, Preprint, 13 pp.
  • [5] Sam Gutmann, J. H. B. Kemperman, J. A. Reeds, and L. A. Shepp, Existence of probability measures with given marginals, Ann. Probab. 19 (1991), no. 4, 1781–1797. MR 1127728
  • [6] Konrad Jacobs, Measure and integral, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. Probability and Mathematical Statistics; With an appendix by Jaroslav Kurzweil. MR 514702
  • [7] Samuel Karlin and William J. Studden, Tchebycheff systems: With applications in analysis and statistics, Pure and Applied Mathematics, Vol. XV, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966. MR 0204922
  • [8] J. H. B. Kemperman, On a class of moment problems, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 101–126. MR 0431335
  • [9] J. H. B. Kemperman, On the role of duality in the theory of moments, Semi-infinite programming and applications (Austin, Tex., 1981) Lecture Notes in Econom. and Math. Systems, vol. 215, Springer, Berlin-New York, 1983, pp. 63–92. MR 709269
  • [10] J. H. B. Kemperman, Geometry of the moment problem, Moments in mathematics (San Antonio, Tex., 1987) Proc. Sympos. Appl. Math., vol. 37, Amer. Math. Soc., Providence, RI, 1987, pp. 16–53. MR 921083, 10.1090/psapm/037/921083
  • [11] J. H. B. Kemperman, Sets of uniqueness and systems of inequalities having a unique solution, Pacific J. Math. 148 (1991), no. 2, 275–301. MR 1094491
  • [12] -, On the lack of uniqueness when reconstructing a set from finitely many central projections (in preparation).
  • [13] J. F. C. Kingman and A. P. Robertson, On a theorem of Lyapunov, J. London Math. Soc. 43 (1968), 347–351. MR 0224768
  • [14] M. G. Kreĭn, The ideas of P. L. Čebyšev and A. A. Markov in the theory of limiting values of integrals and their further development, Amer. Math. Soc. Transl. (2) 12 (1959), 1–121. MR 0113106
  • [15] M. G. Kreĭn and A. A. Nudel′man, The Markov moment problem and extremal problems, American Mathematical Society, Providence, R.I., 1977. Ideas and problems of P. L. Čebyšev and A. A. Markov and their further development; Translated from the Russian by D. Louvish; Translations of Mathematical Monographs, Vol. 50. MR 0458081
  • [16] A. Kuba and A. Volčič, Characterisation of measurable plane sets which are reconstructable from their two projections, Inverse Problems 4 (1988), no. 2, 513–527. MR 954907
  • [17] G. G. Lorentz, A problem of plane measure, Amer. J. Math. 71 (1949), 417–426. MR 0028925
  • [18] A. Volcic, On a class of plane sets, which can be reconstructed from two Steiner symmetrizations, Manuscript.

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Additional Information

DOI: http://dx.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