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Superresolution Rates in Prokhorov Metric

Published online by Cambridge University Press:  20 November 2018

P. Doukhan  and
F. Gamboa
P. Doukhan
Affiliation:
Modélisation stochastique et Statistiques: U.R.A. C.N.R.S. D0743 Orsay Bât 425 F-91405 Orsay Cedex, e-mail: [email protected]
F. Gamboa
Affiliation:
Modélisation stochastique et Statistiques: U.R.A. C.N.R.S D0743 Orsay Bât 425 F-91405 Orsay Cedex and Université Paris-Nord Institut Galilée F-93430 Villetaneuse, e-mail: [email protected]
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Abstract

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Consider the problem of recovering a probability measure supported by a compact subset U of ℝm when the available measurements concern only some of its Ф-moments (Ф being an ℝk valued continuous function on U). When the true Ф-moment c lies on the boundary of the convex hull of Ф(U), generalizing the results of [10], we construct a small set Rα,δ(∊) such that any probability measure μ satisfying is almost concentrated on Rα,δ(∊). When Ф is a pointwise T-system (extension of T-systems), the study of the set Rα,δ(∊) leads to the evaluation of the Prokhorov radius of the set .

Keywords

52A4043A0762A99Generalized momentssuperresolutionT-systemsProkhorov metric
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 2 , 01 April 1996 , pp. 316 - 329
Copyright
Copyright © Canadian Mathematical Society 1996

References

1. Altomare, E. and Campiti, M.,Korovkin-type approximation theory and its applications, Walter de Gruyter-Berlin, 1994.Google Scholar
2. Anastassiou, G.A., The levy radius of a set of probability measures satisfying basic moment conditions involving ﹛t, J2﹜, Constr. Approx, 3 (1987), 257–263.Google Scholar
3., Weak convergence and the Prokhorov radius, J. Math. Anal. Appl. 163 (1992), 541—558.Google Scholar
4. De Vore, R. and Lorentz, G.G.,Constructive Approximation, Springer Verlag, New York, 1993.Google Scholar
5. Donoho, D., Superresolution via sparsity constraints, SIAM J. Math. Anal. 5 (1992), 1309–1331.Google Scholar
6. Donoho, D. and Gassiat, E., Superresolution via positivity constraints, 1992, preprint.Google Scholar
7. Donoho, D., Johnstone, I., Hoch, J. and Stern, , Maximum entropy and the nearly black object, J. Roy. Statist. Soc. Ser. B 54 (1992), 41–82.Google Scholar
8. Gamboa, F. and Gassiat, E., , Math. Programming, Series A 66 (1994), 103–122.Google Scholar
9. Gamboa, F., Bayesian methods for ill posed problems, Ann. Statist. (1996), to appear.Google Scholar
10. Gamboa, F., Sets of superresolution and the maximum entropy method on the mean, SIAM J. Math. Anal., 1996. to appear.Google Scholar
11. Gassiat, E., Probleme sommatoire par maximum d'entropie, C. R. Acad. Sci. Paris Ser. I 303 (1986), 675–680.Google Scholar
12. Gassiat, E., Probleme des moments et concentration de mesure, C. R. Acad. Sci. Paris Ser. I 310 (1990), 41–44.Google Scholar
13. Jansson, P.A.,Deconvolution, With Applications in Spectroscopy, New York, Academic Press, 1984.Google Scholar
14. Karlin, S. and Studden, W.J.,Tchebycheff Systems: With Applications in Analysis and Statistics, John Wiley and Sons, 1966.Google Scholar
15. Krein, M.G. and Nudel'man, A.A., The Markov moment problem and extremal problems, Amer. Math. Soc, 1977.Google Scholar
16. Lewis, A.S., Consistency of moment systems, J. Math. Anal. Appl. (1996), to appear.Google Scholar
17. Lewis, A.S., Superresolution in the Markov moment problem, 1993, preprint.Google Scholar
18. Milnor, J.,Morse theory, Princeton Univ. Press, 1963.Google Scholar
19. Navaza, J., Use ofnon local constraints in maximum entropy electron reconstruction, Acta Cryst. Sect. A 42 (1986), 212–222.Google Scholar
20. Prokhorov, Y.V., Convergence of random processes and limit theorems in probability theory, Theory Probab. Appl. 1 (1956), 157–214.Google Scholar