The injectivity of the Pompeiu transform and \(L^ p\)-analogues of the Wiener-Tauberian theorem. (English) Zbl 0876.43002
D. Pompeiu [Bull. Sci. Math. (2), 53, 328-332 (1929; JFM 55.0138.04)] a associé à toute fonction \(f\in L^1_{\text{loc}}(\mathbb{R}^n)\) et à tout ensemble borné \(E\) de mesure de Lebesgue positive, une fonction \(P_Ef\) définie sur le groupe des déplacements de l’espace affine par \(P_Ef(\sigma)= \int_{\sigma(E)}f\) et a posé le problème de l’injectivité de \(P_E\). Le principal résultat obtenu ici est l’injectivité de \(P_E\) sur \(L^p(\mathbb{R}^n)\) lorsque \(1\leq p\leq2n/(n-1)\).
Reviewer: N.Leblanc (Villetaneuse)
MSC:
| 43A15 | \(L^p\)-spaces and other function spaces on groups, semigroups, etc. |
Keywords:
affine space; transformation group; Pompeiu transform; Wiener-Tauberian theorem; JFM 55.0138.04Citations:
JFM 55.0138.04References:
| [1] | Agranovsky, M.; Berenstein, C.; Chang, D. C., Morera theorem for holomorphic H^p spaces in the Heisenberg group, Journal für die reine und angewandte Mathematik, 443, 49-89 (1993) · Zbl 0781.32009 |
| [2] | Agranovsky, M.; Berenstein, C.; Chang, D. C.; Pascuas, D., Injectivity of the Pompeiu transform in the Heisenberg group, Journal d’Analyse Mathématique, 63, 131-173 (1994) · Zbl 0808.43002 · doi:10.1007/BF03008422 |
| [3] | Bagchi, S. C.; Sitaram, A., The Pompeiu problem revisited, L’Enseignement Mathématique, 36, 67-91 (1990) · Zbl 0722.43009 |
| [4] | Berenstein, C. A.; Zalcman, L., Pompeiu’s problem on symmetric spaces, Commentarii Mathematici Helvetici, 55, 593-621 (1980) · Zbl 0452.43012 · doi:10.1007/BF02566709 |
| [5] | Brown, L.; Schnitzer, F.; Shields, A. L., A note on a problem of D. Pompeiu, Mathematische Zeitschrift, 105, 59-61 (1968) · Zbl 0161.24704 · doi:10.1007/BF01135449 |
| [6] | Brown, L.; Schreiber, B. M.; Taylor, B. A., Spectral synthesis and the Pompeiu problem, Annales de l’Institut Fourier (Grenoble), 23, 125-154 (1973) · Zbl 0265.46044 |
| [7] | Rudin, W., Functional Analysis (1973), New York: McGraw-Hill, New York · Zbl 0253.46001 |
| [8] | Shahshahani, M.; Sitaram, A., The Pompeiu problem in exterior domains in symmetric spaces, Comtemporary Mathematics, 63, 267-277 (1987) · Zbl 0615.53060 |
| [9] | Sitaram, A., Fourier analysis and determining sets for Radon measures on ℝ^n, Illinois Journal of Mathematics, 28, 339-347 (1984) · Zbl 0581.28008 |
| [10] | Sitaram, A., Some remarks on measures on noncompact semisimple Lie groups, Pacific Journal of Mathematics, 110, 429-434 (1984) · Zbl 0532.43001 |
| [11] | Sitaram, A., On an analogue of the Wiener-Tauberian theorem for symmetric spaces of the non-compact type, Pacific Journal of Mathematics, 133, 197-208 (1988) · Zbl 0628.43011 |
| [12] | Thangavelu, S., Spherical means and CR functions on the Heisenberg group, Journal d’Analyse Mathématique, 63, 255-286 (1994) · Zbl 0822.43001 |
| [13] | Zalcman, L., Offbeat integral geometry, The American Mathematical Monthly, 87, 161-175 (1980) · Zbl 0433.53048 · doi:10.2307/2321600 |
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