# American Institute of Mathematical Sciences

November  2010, 4(4): 619-630. doi: 10.3934/ipi.2010.4.619

## Unique continuation of microlocally analytic distributions and injectivity theorems for the ray transform

 1 Department of Mathematics, Stockholm University, SE-10691 Stockholm

Received  March 2009 Published  September 2010

Using a vanishing theorem for microlocally real analytic distributions and a theorem on flatness of a distribution vanishing on infinitely many hyperplanes we give a new proof of an injectivity theorem of Bélisle, Massé, and Ransford for the ray transform on $\R^n$. By means of an example we show that this result is sharp. An extension is given where real analyticity is replaced by quasianalyticity.
Citation: Jan Boman. Unique continuation of microlocally analytic distributions and injectivity theorems for the ray transform. Inverse Problems and Imaging, 2010, 4 (4) : 619-630. doi: 10.3934/ipi.2010.4.619
##### References:
 [1] C. Béslisle, J.-C. Massé and T. Ransford, When is a probability measure determined by infinitely many projections?, Ann. Probab., 25 (1997), 767-786. doi: doi:10.1214/aop/1024404418. [2] J. Boman, A local vanishing theorem for distributions, C. R. Acad. Sci. Paris, Série I, 315 (1992), 1231-1234. [3] J. Boman, Microlocal quasianalyticity for distributions and ultradistributions, Publ. Res. Inst. Math. Sci. (Kyoto), 31 (1995), 1079-1095. doi: doi:10.2977/prims/1195163598. [4] J. Boman, Flatness of distributions vanishing on infinitely many hyperplanes, C. R. Acad. Sci. Paris, Série I, 347 (2009), 1351-1354. [5] L. Hörmander, Uniqueness theorems and wave front sets for solutions of linear differential equations with analytic coefficients, Comm. Pure Appl. Math., 24 (1971), 671-704. doi: doi:10.1002/cpa.3160240505. [6] L. Hörmander, "The Analysis of Linear Partial Differential Operators," Vol. 1, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983. [7] L. Hörmander, Remarks on Holmgren's uniqueness theorem, Ann. Inst. Fourier (Grenoble), 43 (1993), 1223-1251. [8] D. Iagolnitzer, Appendix: Microlocal essential support of a distribution and decomposition theorems-An introduction, in "Hyperfunctions and Theoretical Physics," Lecture Notes in Math., 449 (1975), 121-132. [9] F. Natterer, "The Mathematics of Computerized Tomography," Wiley&Sons, New York, Brisbane, Toronto, Singapore, 1986. [10] F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction," SIAM, Philadelphia, 2001. [11] V. Palamodov, "Reconstructive Integral Geometry," Birkhäuser, Basel, Boston, Berlin, 2004.

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##### References:
 [1] C. Béslisle, J.-C. Massé and T. Ransford, When is a probability measure determined by infinitely many projections?, Ann. Probab., 25 (1997), 767-786. doi: doi:10.1214/aop/1024404418. [2] J. Boman, A local vanishing theorem for distributions, C. R. Acad. Sci. Paris, Série I, 315 (1992), 1231-1234. [3] J. Boman, Microlocal quasianalyticity for distributions and ultradistributions, Publ. Res. Inst. Math. Sci. (Kyoto), 31 (1995), 1079-1095. doi: doi:10.2977/prims/1195163598. [4] J. Boman, Flatness of distributions vanishing on infinitely many hyperplanes, C. R. Acad. Sci. Paris, Série I, 347 (2009), 1351-1354. [5] L. Hörmander, Uniqueness theorems and wave front sets for solutions of linear differential equations with analytic coefficients, Comm. Pure Appl. Math., 24 (1971), 671-704. doi: doi:10.1002/cpa.3160240505. [6] L. Hörmander, "The Analysis of Linear Partial Differential Operators," Vol. 1, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983. [7] L. Hörmander, Remarks on Holmgren's uniqueness theorem, Ann. Inst. Fourier (Grenoble), 43 (1993), 1223-1251. [8] D. Iagolnitzer, Appendix: Microlocal essential support of a distribution and decomposition theorems-An introduction, in "Hyperfunctions and Theoretical Physics," Lecture Notes in Math., 449 (1975), 121-132. [9] F. Natterer, "The Mathematics of Computerized Tomography," Wiley&Sons, New York, Brisbane, Toronto, Singapore, 1986. [10] F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction," SIAM, Philadelphia, 2001. [11] V. Palamodov, "Reconstructive Integral Geometry," Birkhäuser, Basel, Boston, Berlin, 2004.
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