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Ray transforms on a conformal class of curves
1. | University of Toronto, Department of Mathematics, Toronto, ON, M5S 2E4, Canada |
2. | Columbia University, Department of Applied Physics and Applied Mathematics, New York, NY, 10025, United States |
References:
[1] | |
[2] |
L. V. Ahlfors, Lectures on Quasiconformal Mappings, University Lecture Series, American Mathematical Society, 2006. |
[3] |
L. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics, Ann. of Math, 72 (1960), 385-404. |
[4] |
E. Arbuzov, A. Bukhgeim and S. Kazantsev, Two-dimensional tomography problems and the theory of A-analytic functions, Siberian Advances in Mathematics, 8 (1998), 1-20. |
[5] |
K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton University Press, 2009. |
[6] |
G. Bal, Ray transforms in hyperbolic geometry, J. Math. Pures Appl., 84 (2005), 1362-1392.
doi: 10.1016/j.matpur.2005.02.001. |
[7] |
G. Bal, On the attenuated Radon transform with full and partial measurements, Inverse Problems, 20 (2004), 399-418.
doi: 10.1088/0266-5611/20/2/006. |
[8] |
H. Begehr, Complex Analytic Methods for Partial Differential Equations, World Scientific Publishing Co., 1994. |
[9] |
C. Berenstein and E. C. Tarabush, Integral geometry in hyperbolic spaces and electrical impedance tomography, SIAM J. Appl. Math., 56 (1996), 755-764.
doi: 10.1137/S0036139994277348. |
[10] |
P. Colwell, Blaschke Products: Bounded Analytical Functions, University of Michigan Press, 1985. |
[11] |
L. Ehrenpreis, The Universality of the Radon Transform, Oxford Mathematical Monographs, Oxford University Press, USA, 2003.
doi: 10.1093/acprof:oso/9780198509783.001.0001. |
[12] |
L. C. Evans, Partial Differential Equations, 19 of Graduate Studies in Mathematics, American Mathematical Society, 1998. |
[13] |
D. Finch, Uniqueness for the X-ray transform in the physical range, Inverse Problems, 2 (1986), 197-203.
doi: 10.1088/0266-5611/2/2/010. |
[14] |
M. J. Ablowitz and A. S. Fokas, Complex Variables: Introduction and Applications, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2003.
doi: 10.1017/CBO9780511791246. |
[15] |
J. B. Garnett, Bounded Analytic Functions, Springer New York, 1981. |
[16] |
R. E. Greene and S. G. Krantz, Function Theory of One Complex Variable, 40 of Graduate Studies in Mathematics, American Mathematical Society, 2006. |
[17] |
D. Griffiths, Introduction to Elementary Particles, Wiley-VCH, 2008.
doi: 10.1002/9783527618460. |
[18] |
S. Helgason, The Radon Transform, 5 of Progress in Mathematics, Birkhäuser Boston, 1980. |
[19] |
______, Groups and Geometric Analysis (Integral Geometry, Invariant Differential Operators and Spherical Functions), American Mathematical Society, 2000. |
[20] |
______, The inversion of the x-ray transform on a compact symmetric space, Journal of Lie Theory, 17 (2007), 307-315. |
[21] |
L. Hormander, Complex Analysis in Several Variables, North Holland, 1990. |
[22] |
S. S. Romesh Kumar, Inner functions and substitution operators, Acta Sci. Math. (Szegal), 58 (1993), 509-516. |
[23] |
J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, 176 in Graduate Texts in Mathematics, Springer, 1997. |
[24] |
F. Natterer, Inversion of the attenuated radon transform, Inverse Problems, 17 (2001), 113-119.
doi: 10.1088/0266-5611/17/1/309. |
[25] |
F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, (Monographs on Mathematical Modeling and Computation), Society for Industrial Mathematics, 2007.
doi: 10.1118/1.1455744. |
[26] |
Z. Nehari, Conformal Mappings, McGraw-Hill Book Company, Inc., 1952. |
[27] |
R. Novikov, An inversion formula for the attenuated x-ray transformation, Ark. Math, 40 (2002), 145-167.
doi: 10.1007/BF02384507. |
[28] |
L. Pestov and G. Uhlmann, On characterization of range and inversion formulas for the geodesic x-ray transform, International Math. Research Notices, 80 (2004), 4331-4347.
doi: 10.1155/S1073792804142116. |
[29] |
H. Renelt, Elliptic Systems and Quasiconformal Mappings, John Wiley & Sons Inc, 1988. |
[30] |
V. Rubakov and S. S. Wilson, Classical Theory of Gauge Fields, Princeton University Press, 2002. |
[31] |
B. Rubin, Notes on radon transforms in integral geometry, Fract. Calc. Appl. Anal., 6 (2003), 25-72. |
[32] |
M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces, J. Diff. Geom., 88 (2011), 161-187. |
[33] |
D. Sarason, Complex Function Theory, American Mathematical Society, second ed., 2007. |
[34] |
V. Sharafudtinov, Integral Geometry of Tensor Fields, VSP, Utrecht, The Netherlands, 1994.
doi: 10.1515/9783110900095. |
[35] |
G. Uhlmann, Inside Out: Inverse Problems and Applications, Cambridge University Press, 2003.
doi: 10.1090/conm/333. |
[36] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. |
[37] |
M. E. Taylor, Partial Differential Equations, vol. 115-117 of Applied Mathematical Sciences, Springer, 1996. |
show all references
References:
[1] | |
[2] |
L. V. Ahlfors, Lectures on Quasiconformal Mappings, University Lecture Series, American Mathematical Society, 2006. |
[3] |
L. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics, Ann. of Math, 72 (1960), 385-404. |
[4] |
E. Arbuzov, A. Bukhgeim and S. Kazantsev, Two-dimensional tomography problems and the theory of A-analytic functions, Siberian Advances in Mathematics, 8 (1998), 1-20. |
[5] |
K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton University Press, 2009. |
[6] |
G. Bal, Ray transforms in hyperbolic geometry, J. Math. Pures Appl., 84 (2005), 1362-1392.
doi: 10.1016/j.matpur.2005.02.001. |
[7] |
G. Bal, On the attenuated Radon transform with full and partial measurements, Inverse Problems, 20 (2004), 399-418.
doi: 10.1088/0266-5611/20/2/006. |
[8] |
H. Begehr, Complex Analytic Methods for Partial Differential Equations, World Scientific Publishing Co., 1994. |
[9] |
C. Berenstein and E. C. Tarabush, Integral geometry in hyperbolic spaces and electrical impedance tomography, SIAM J. Appl. Math., 56 (1996), 755-764.
doi: 10.1137/S0036139994277348. |
[10] |
P. Colwell, Blaschke Products: Bounded Analytical Functions, University of Michigan Press, 1985. |
[11] |
L. Ehrenpreis, The Universality of the Radon Transform, Oxford Mathematical Monographs, Oxford University Press, USA, 2003.
doi: 10.1093/acprof:oso/9780198509783.001.0001. |
[12] |
L. C. Evans, Partial Differential Equations, 19 of Graduate Studies in Mathematics, American Mathematical Society, 1998. |
[13] |
D. Finch, Uniqueness for the X-ray transform in the physical range, Inverse Problems, 2 (1986), 197-203.
doi: 10.1088/0266-5611/2/2/010. |
[14] |
M. J. Ablowitz and A. S. Fokas, Complex Variables: Introduction and Applications, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2003.
doi: 10.1017/CBO9780511791246. |
[15] |
J. B. Garnett, Bounded Analytic Functions, Springer New York, 1981. |
[16] |
R. E. Greene and S. G. Krantz, Function Theory of One Complex Variable, 40 of Graduate Studies in Mathematics, American Mathematical Society, 2006. |
[17] |
D. Griffiths, Introduction to Elementary Particles, Wiley-VCH, 2008.
doi: 10.1002/9783527618460. |
[18] |
S. Helgason, The Radon Transform, 5 of Progress in Mathematics, Birkhäuser Boston, 1980. |
[19] |
______, Groups and Geometric Analysis (Integral Geometry, Invariant Differential Operators and Spherical Functions), American Mathematical Society, 2000. |
[20] |
______, The inversion of the x-ray transform on a compact symmetric space, Journal of Lie Theory, 17 (2007), 307-315. |
[21] |
L. Hormander, Complex Analysis in Several Variables, North Holland, 1990. |
[22] |
S. S. Romesh Kumar, Inner functions and substitution operators, Acta Sci. Math. (Szegal), 58 (1993), 509-516. |
[23] |
J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, 176 in Graduate Texts in Mathematics, Springer, 1997. |
[24] |
F. Natterer, Inversion of the attenuated radon transform, Inverse Problems, 17 (2001), 113-119.
doi: 10.1088/0266-5611/17/1/309. |
[25] |
F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, (Monographs on Mathematical Modeling and Computation), Society for Industrial Mathematics, 2007.
doi: 10.1118/1.1455744. |
[26] |
Z. Nehari, Conformal Mappings, McGraw-Hill Book Company, Inc., 1952. |
[27] |
R. Novikov, An inversion formula for the attenuated x-ray transformation, Ark. Math, 40 (2002), 145-167.
doi: 10.1007/BF02384507. |
[28] |
L. Pestov and G. Uhlmann, On characterization of range and inversion formulas for the geodesic x-ray transform, International Math. Research Notices, 80 (2004), 4331-4347.
doi: 10.1155/S1073792804142116. |
[29] |
H. Renelt, Elliptic Systems and Quasiconformal Mappings, John Wiley & Sons Inc, 1988. |
[30] |
V. Rubakov and S. S. Wilson, Classical Theory of Gauge Fields, Princeton University Press, 2002. |
[31] |
B. Rubin, Notes on radon transforms in integral geometry, Fract. Calc. Appl. Anal., 6 (2003), 25-72. |
[32] |
M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces, J. Diff. Geom., 88 (2011), 161-187. |
[33] |
D. Sarason, Complex Function Theory, American Mathematical Society, second ed., 2007. |
[34] |
V. Sharafudtinov, Integral Geometry of Tensor Fields, VSP, Utrecht, The Netherlands, 1994.
doi: 10.1515/9783110900095. |
[35] |
G. Uhlmann, Inside Out: Inverse Problems and Applications, Cambridge University Press, 2003.
doi: 10.1090/conm/333. |
[36] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. |
[37] |
M. E. Taylor, Partial Differential Equations, vol. 115-117 of Applied Mathematical Sciences, Springer, 1996. |
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