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The Pompeiu problem for groups

Published online by Cambridge University Press:  24 October 2008

Alan L. Carey ,
Eberhard Kaniuth  and
William Moran
Alan L. Carey
Affiliation:
Department of Pure Mathematics, University of Adelaide, Adelaide Sit 5001, Australia
Eberhard Kaniuth
Affiliation:
Fachbereich Mathematik/informatik, Universität Paderborn, D-4790 Paderborn, Federal Republic of Germany
William Moran
Affiliation:
Department of Pure Mathematics, University of Adelaide, Adelaide SA 5001, Australia
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Extract

The Pompeiu problem has its origins in classical analysis in ℝn (see [2, 3, 4, 8] for a discussion and some history). In this context it may be stated as follows. Let D ⊂ ℝn be a bounded measurable set of positive Lebesgue measure and f a locally integrable function on ℝn. Then, if ∫σ(D)f = 0 for all rigid motions σ of ℝn, is f = 0?

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 109 , Issue 1 , January 1991 , pp. 45 - 58
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

[1]Bagchi, S. C. and Sitabam, A.. Determining sets for measures on ℝn. Illinoìs J. Math. 26 (1982), 419–422.Google Scholar
[2]Berenstein, C. A.. An inverse spectral theorem and its relation to the Pompeiu problem. J. Analyse Math. 37 (1980), 128–144.CrossRefGoogle Scholar
[3]Berenstein, C. A. and Shahshahani, M.. Harmonic analysis and the Pompeiu problem. Amer. J.Math. 105 (1983), 1217–1229.CrossRefGoogle Scholar
[4]Berenstein, C. A. and Zalcman, L.. The Pompeiu problem in symmetric spaces. Comment. Math. Helv. 55 (1980), 593–621.Google Scholar
[5]Blattner, R. J.. On induced representations. Amer. J. Math. 83 (1961), 79–98.CrossRefGoogle Scholar
[6]Boidol, J., Leptin, H., Schürmann, F. and Vahle, D.. Räume primitiver Ideale von Gruppenalgebren. Math. Ann. 236 (1978), 1–13.CrossRefGoogle Scholar
[7]Bredon, G. E.. Introduction to Compact Transformation Groups (Academic Press, 1972).Google Scholar
[8]Brown, L., Schreiber, B. M. and Taylor, B. A.. Spectral synthesis and the Pompeiu problem. Ann. Inst. Fourier (Grenoble) 23 (1973), 125–154.CrossRefGoogle Scholar
[9]Dixmier, J.. Les C*-algèbres et leurs Représentations (Gauthier-Villars, 1964).Google Scholar
[10]Feldman, J. and Greenleaf, F. P.. Existence of Borel transversals in groups. Pacific J. Math. 25 (1968), 455–461.CrossRefGoogle Scholar
[11]Fell, J. M. G.. Weak containment and induced representations. II. Trans. Amer. Math. Soc. 110 (1964), 424–447.Google Scholar
[12]Fell, J. M. G.. A Hausdorff topology on the closed subsets of a locally compact non-Hausdorff space. Proc. Amer. Math. Soc. 13 (1962), 472–476.CrossRefGoogle Scholar
[13]Glimm, J.. Locally compact transformation groups. Trans. Amer. Math. Soc. 101 (1961), 124–138.CrossRefGoogle Scholar
[14]Gluškov, V. M.. Locally nilpotent locally bicompact groups (in Russian). Trudy Moskov. Mat. Obshch. 4 (1955), 291–332.Google Scholar
[15]Green, P.. The local structure of twisted covariance algebras. Acta Math. 140 (1978), 191–250.CrossRefGoogle Scholar
[16]Grosser, S. and Moskowitz, M.. Compactness conditions in topological groups. J. Reine Angew.Math. 246 (1971), 1–40.Google Scholar
[17]Hauenschild, W.. Zur Darstellungstheorie von SIN-Gruppen. Math. Ann. 210 (1974), 257–276.CrossRefGoogle Scholar
[18]Hochschild, G.. The Structure of Lie Groups (Holden-Day, 1965).Google Scholar
[19]Kaniuth, E.. On maximal ideals in group algebras of SIN-groups. Math. Ann. 214 (1975), 167–175.CrossRefGoogle Scholar
[20]Ludwig, J.. Good ideals in the group algebra of a nilpotent Lie group. Math. Z. 161 (1978), 195–210.CrossRefGoogle Scholar
[21]Mosak, R.. The L 1- and C*-algebras of -groups and their representations. Trans. Amer. Math. Soc. 163 (1972), 277–310.Google Scholar
[22]Rana, I. K.. Determination of probability measures through group actions. Z. Wahrsch. Verw. Gebiete 53 (1980), 197–206.CrossRefGoogle Scholar
[23]Rieffel, M. A.. Applications of strong Morita equivalence to transformation group C*-algebras. Proc. Symposia Pure Math. 38 (1982), 299–309.CrossRefGoogle Scholar
[24]Scott, D. and Sitaram, A.. Some remarks on the Pompeiu problem for groups. Proc. Amer. Math. Soc. 104 (1988), 1261–1266.CrossRefGoogle Scholar
[25]Thoma, E.. Über das reguläre Mass im dualen Raum diskreter Gruppen. Math. Z. 100 (1967), 257–271.CrossRefGoogle Scholar
[26]Weit, Y.. On P-sets for group algebras of semi-direct products of abelian groups. (Preprint.)Google Scholar
[27]Williams, D. P.. The topology on the primitive ideal space of transformation group C*-algebras. Trans. Amer. Math. Soc. 266 (1981), 335–359.Google Scholar
[28]Zalcman, L.. Offbeat integral geometry. Amer. Math. Monthly 87 (1980), 161–175.CrossRefGoogle Scholar