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 & 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. doi: doi:10.1214/aop/1024404418.

[2]

J. Boman, A local vanishing theorem for distributions,, C. R. Acad. Sci. Paris, 315 (1992), 1231.

[3]

J. Boman, Microlocal quasianalyticity for distributions and ultradistributions,, Publ. Res. Inst. Math. Sci. (Kyoto), 31 (1995), 1079. doi: doi:10.2977/prims/1195163598.

[4]

J. Boman, Flatness of distributions vanishing on infinitely many hyperplanes,, C. R. Acad. Sci. Paris, 347 (2009), 1351.

[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. doi: doi:10.1002/cpa.3160240505.

[6]

L. Hörmander, "The Analysis of Linear Partial Differential Operators," Vol. 1,, Springer-Verlag, (1983).

[7]

L. Hörmander, Remarks on Holmgren's uniqueness theorem,, Ann. Inst. Fourier (Grenoble), 43 (1993), 1223.

[8]

D. Iagolnitzer, Appendix: Microlocal essential support of a distribution and decomposition theorems-An introduction,, in, 449 (1975), 121.

[9]

F. Natterer, "The Mathematics of Computerized Tomography,", Wiley&Sons, (1986).

[10]

F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction,", SIAM, (2001).

[11]

V. Palamodov, "Reconstructive Integral Geometry,", Birkhäuser, (2004).

show all references

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. doi: doi:10.1214/aop/1024404418.

[2]

J. Boman, A local vanishing theorem for distributions,, C. R. Acad. Sci. Paris, 315 (1992), 1231.

[3]

J. Boman, Microlocal quasianalyticity for distributions and ultradistributions,, Publ. Res. Inst. Math. Sci. (Kyoto), 31 (1995), 1079. doi: doi:10.2977/prims/1195163598.

[4]

J. Boman, Flatness of distributions vanishing on infinitely many hyperplanes,, C. R. Acad. Sci. Paris, 347 (2009), 1351.

[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. doi: doi:10.1002/cpa.3160240505.

[6]

L. Hörmander, "The Analysis of Linear Partial Differential Operators," Vol. 1,, Springer-Verlag, (1983).

[7]

L. Hörmander, Remarks on Holmgren's uniqueness theorem,, Ann. Inst. Fourier (Grenoble), 43 (1993), 1223.

[8]

D. Iagolnitzer, Appendix: Microlocal essential support of a distribution and decomposition theorems-An introduction,, in, 449 (1975), 121.

[9]

F. Natterer, "The Mathematics of Computerized Tomography,", Wiley&Sons, (1986).

[10]

F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction,", SIAM, (2001).

[11]

V. Palamodov, "Reconstructive Integral Geometry,", Birkhäuser, (2004).

[1]

Artem Dudko. Computability of the Julia set. Nonrecurrent critical orbits. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2751-2778. doi: 10.3934/dcds.2014.34.2751

[2]

Gareth Ainsworth. The attenuated magnetic ray transform on surfaces. Inverse Problems & Imaging, 2013, 7 (1) : 27-46. doi: 10.3934/ipi.2013.7.27

[3]

Gareth Ainsworth. The magnetic ray transform on Anosov surfaces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1801-1816. doi: 10.3934/dcds.2015.35.1801

[4]

Dan Jane, Gabriel P. Paternain. On the injectivity of the X-ray transform for Anosov thermostats. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 471-487. doi: 10.3934/dcds.2009.24.471

[5]

Yiran Wang. Parametrices for the light ray transform on Minkowski spacetime. Inverse Problems & Imaging, 2018, 12 (1) : 229-237. doi: 10.3934/ipi.2018009

[6]

Gareth Ainsworth, Yernat M. Assylbekov. On the range of the attenuated magnetic ray transform for connections and Higgs fields. Inverse Problems & Imaging, 2015, 9 (2) : 317-335. doi: 10.3934/ipi.2015.9.317

[7]

François Rouvière. X-ray transform on Damek-Ricci spaces. Inverse Problems & Imaging, 2010, 4 (4) : 713-720. doi: 10.3934/ipi.2010.4.713

[8]

Venkateswaran P. Krishnan, Plamen Stefanov. A support theorem for the geodesic ray transform of symmetric tensor fields. Inverse Problems & Imaging, 2009, 3 (3) : 453-464. doi: 10.3934/ipi.2009.3.453

[9]

Mark Hubenthal. The broken ray transform in $n$ dimensions with flat reflecting boundary. Inverse Problems & Imaging, 2015, 9 (1) : 143-161. doi: 10.3934/ipi.2015.9.143

[10]

Siamak RabieniaHaratbar. Support theorem for the Light-Ray transform of vector fields on Minkowski spaces. Inverse Problems & Imaging, 2018, 12 (2) : 293-314. doi: 10.3934/ipi.2018013

[11]

Jong-Shenq Guo, Hirokazu Ninomiya, Chin-Chin Wu. Existence of a rotating wave pattern in a disk for a wave front interaction model. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1049-1063. doi: 10.3934/cpaa.2013.12.1049

[12]

Marcelo Disconzi, Daniel Toundykov, Justin T. Webster. Front matter. Evolution Equations & Control Theory, 2016, 5 (4) : i-iii. doi: 10.3934/eect.201604i

[13]

Y. A. Li, P. J. Olver. Convergence of solitary-wave solutions in a perturbed bi-hamiltonian dynamical system ii. complex analytic behavior and convergence to non-analytic solutions. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 159-191. doi: 10.3934/dcds.1998.4.159

[14]

Nicholas Hoell, Guillaume Bal. Ray transforms on a conformal class of curves. Inverse Problems & Imaging, 2014, 8 (1) : 103-125. doi: 10.3934/ipi.2014.8.103

[15]

Debora Amadori, Wen Shen. Front tracking approximations for slow erosion. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1481-1502. doi: 10.3934/dcds.2012.32.1481

[16]

Jaume Llibre, Claudia Valls. On the analytic integrability of the Liénard analytic differential systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 557-573. doi: 10.3934/dcdsb.2016.21.557

[17]

Doǧan Çömez. The modulated ergodic Hilbert transform. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 325-336. doi: 10.3934/dcdss.2009.2.325

[18]

Sean Holman, Plamen Stefanov. The weighted Doppler transform. Inverse Problems & Imaging, 2010, 4 (1) : 111-130. doi: 10.3934/ipi.2010.4.111

[19]

Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006

[20]

David W. Pravica, Michael J. Spurr. Analytic continuation into the future. Conference Publications, 2003, 2003 (Special) : 709-716. doi: 10.3934/proc.2003.2003.709

2017 Impact Factor: 1.465

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]