American Institute of Mathematical Sciences

Advanced
May  2013, 7(2): 491-498. doi: 10.3934/ipi.2013.7.491

A note on analyticity properties of far field patterns

Roland Griesmaier 1, , Nuutti Hyvönen 2, and  Otto Seiskari 3,

1. 

Mathematisches Institut, Universität Leipzig, 04009 Leipzig, Germany
2. 

Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, FI-00076 Aalto
3. 

Aalto University, Department of Mathematics and Systems Analysis, FI-00076 Aalto, Finland

Received  October 2012 Revised  December 2012 Published  May 2013

In scattering theory the far field pattern describes the directional dependence of a time-harmonic wave scattered by an obstacle or inhomogeneous medium, when observed sufficiently far away from these objects. Considering plane wave excitations, the far field pattern can be written as a function of two variables, namely the direction of propagation of the incident plane wave and the observation direction, and it is well-known to be separately real analytic with respect to each of them. We show that the far field pattern is in fact a jointly real analytic function of these two variables.
Keywords: partial data., Far field pattern, inverse scattering, analytic functions.
Mathematics Subject Classification: 35B30, 35R30, 35Q6.
Citation: Roland Griesmaier, Nuutti Hyvönen, Otto Seiskari. A note on analyticity properties of far field patterns. Inverse Problems & Imaging, 2013, 7 (2) : 491-498. doi: 10.3934/ipi.2013.7.491
References:
[1]

F. E. Browder, Real analytic functions on product spaces and separate analyticity, Canad. J. Math., 13 (1961), 650-656. doi: 10.4153/CJM-1961-054-1.  Google Scholar

[2]

A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16 (2008), 19-33. doi: 10.1515/jiip.2008.002.  Google Scholar

[3]

D. L. Colton and R. Kress, "Integral Equation Methods in Scattering Theory," John Wiley & Sons, New York, 1983.  Google Scholar

[4]

D. L. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," $2^{nd}$ edition, Springer-Verlag, Berlin, 1998.  Google Scholar

[5]

F. Hartogs, Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten, Math. Ann., 62 (1906), 1-88. doi: 10.1007/BF01448415.  Google Scholar

[6]

L. Hörmander, "An Introduction to Complex Analysis in Several Variables," $3^{rd}$ edition, North-Holland, Amsterdam, 1990.  Google Scholar

[7]

N. Hyvönen, P. Piiroinen and O. Seiskari, Point measurements for a Neumann-to-Dirichlet map and the Calderón problem in the plane, SIAM J. Math. Anal., 44 (2012), 3526-3536. doi: 10.1137/120872164.  Google Scholar

[8]

A. Kirsch, The MUSIC algorithm and the factorization method in inverse scattering theory for inhomogeneous media, Inverse Problems, 18 (2002), 1025-1040. doi: 10.1088/0266-5611/18/4/306.  Google Scholar

[9]

A. Kirsch, "An Introduction to the Mathematical Theory of Inverse Problems," $2^{nd}$ edition, Springer-Verlag, New York, 2011. doi: 10.1007/978-1-4419-8474-6.  Google Scholar

[10]

A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford University Press, Oxford, 2008.  Google Scholar

[11]

S. G. Krantz, "Function Theory of Several Complex Variables," $2^{nd}$ edition, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992.  Google Scholar

[12]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications," I, Springer-Verlag, New York, 1972.  Google Scholar

[13]

A. I. Nachman, Reconstructions from boundary measurements, Ann. of Math., 128 (1988), 531-576. doi: 10.2307/1971435.  Google Scholar

[14]

R. G. Novikov, A multidimensional inverse spectral problem for the equation $-\Delta \psi + (v(x)-Eu(x))\psi=0$, translation in Funct. Anal. Appl., 22 (1988), 263-272. doi: 10.1007/BF01077418.  Google Scholar

[15]

A. G. Ramm, Recovery of the potential from fixed-energy scattering data, Inverse Problems, 4 (1988), 877-886. doi: 10.1088/0266-5611/4/3/020.  Google Scholar

show all references

References:
[1]

F. E. Browder, Real analytic functions on product spaces and separate analyticity, Canad. J. Math., 13 (1961), 650-656. doi: 10.4153/CJM-1961-054-1.  Google Scholar

[2]

A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16 (2008), 19-33. doi: 10.1515/jiip.2008.002.  Google Scholar

[3]

D. L. Colton and R. Kress, "Integral Equation Methods in Scattering Theory," John Wiley & Sons, New York, 1983.  Google Scholar

[4]

D. L. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," $2^{nd}$ edition, Springer-Verlag, Berlin, 1998.  Google Scholar

[5]

F. Hartogs, Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten, Math. Ann., 62 (1906), 1-88. doi: 10.1007/BF01448415.  Google Scholar

[6]

L. Hörmander, "An Introduction to Complex Analysis in Several Variables," $3^{rd}$ edition, North-Holland, Amsterdam, 1990.  Google Scholar

[7]

N. Hyvönen, P. Piiroinen and O. Seiskari, Point measurements for a Neumann-to-Dirichlet map and the Calderón problem in the plane, SIAM J. Math. Anal., 44 (2012), 3526-3536. doi: 10.1137/120872164.  Google Scholar

[8]

A. Kirsch, The MUSIC algorithm and the factorization method in inverse scattering theory for inhomogeneous media, Inverse Problems, 18 (2002), 1025-1040. doi: 10.1088/0266-5611/18/4/306.  Google Scholar

[9]

A. Kirsch, "An Introduction to the Mathematical Theory of Inverse Problems," $2^{nd}$ edition, Springer-Verlag, New York, 2011. doi: 10.1007/978-1-4419-8474-6.  Google Scholar

[10]

A. Kirsch and N. Grinberg, "The Factorization Method for Inverse Problems," Oxford University Press, Oxford, 2008.  Google Scholar

[11]

S. G. Krantz, "Function Theory of Several Complex Variables," $2^{nd}$ edition, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992.  Google Scholar

[12]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications," I, Springer-Verlag, New York, 1972.  Google Scholar

[13]

A. I. Nachman, Reconstructions from boundary measurements, Ann. of Math., 128 (1988), 531-576. doi: 10.2307/1971435.  Google Scholar

[14]

R. G. Novikov, A multidimensional inverse spectral problem for the equation $-\Delta \psi + (v(x)-Eu(x))\psi=0$, translation in Funct. Anal. Appl., 22 (1988), 263-272. doi: 10.1007/BF01077418.  Google Scholar

[15]

A. G. Ramm, Recovery of the potential from fixed-energy scattering data, Inverse Problems, 4 (1988), 877-886. doi: 10.1088/0266-5611/4/3/020.  Google Scholar

[1]

Huai-An Diao, Peijun Li, Xiaokai Yuan. Inverse elastic surface scattering with far-field data. Inverse Problems & Imaging, 2019, 13 (4) : 721-744. doi: 10.3934/ipi.2019033
[2]

Jingzhi Li, Jun Zou. A direct sampling method for inverse scattering using far-field data. Inverse Problems & Imaging, 2013, 7 (3) : 757-775. doi: 10.3934/ipi.2013.7.757
[3]

Olha Ivanyshyn. Shape reconstruction of acoustic obstacles from the modulus of the far field pattern. Inverse Problems & Imaging, 2007, 1 (4) : 609-622. doi: 10.3934/ipi.2007.1.609
[4]

Xiaoxu Xu, Bo Zhang, Haiwen Zhang. Uniqueness in inverse acoustic and electromagnetic scattering with phaseless near-field data at a fixed frequency. Inverse Problems & Imaging, 2020, 14 (3) : 489-510. doi: 10.3934/ipi.2020023
[5]

Francesco Demontis, Cornelis Van der Mee. Novel formulation of inverse scattering and characterization of scattering data. Conference Publications, 2011, 2011 (Special) : 343-350. doi: 10.3934/proc.2011.2011.343
[6]

Giovanni Alessandrini, Eva Sincich, Sergio Vessella. Stable determination of surface impedance on a rough obstacle by far field data. Inverse Problems & Imaging, 2013, 7 (2) : 341-351. doi: 10.3934/ipi.2013.7.341
[7]

Qi Wang, Yanren Hou. Determining an obstacle by far-field data measured at a few spots. Inverse Problems & Imaging, 2015, 9 (2) : 591-600. doi: 10.3934/ipi.2015.9.591
[8]

Xiaosheng Li, Gunther Uhlmann. Inverse problems with partial data in a slab. Inverse Problems & Imaging, 2010, 4 (3) : 449-462. doi: 10.3934/ipi.2010.4.449
[9]

Miklós Horváth. Spectral shift functions in the fixed energy inverse scattering. Inverse Problems & Imaging, 2011, 5 (4) : 843-858. doi: 10.3934/ipi.2011.5.843
[10]

Masaru Ikehata, Esa Niemi, Samuli Siltanen. Inverse obstacle scattering with limited-aperture data. Inverse Problems & Imaging, 2012, 6 (1) : 77-94. doi: 10.3934/ipi.2012.6.77
[11]

Hanming Zhou. Lens rigidity with partial data in the presence of a magnetic field. Inverse Problems & Imaging, 2018, 12 (6) : 1365-1387. doi: 10.3934/ipi.2018057
[12]

Heping Dong, Deyue Zhang, Yukun Guo. A reference ball based iterative algorithm for imaging acoustic obstacle from phaseless far-field data. Inverse Problems & Imaging, 2019, 13 (1) : 177-195. doi: 10.3934/ipi.2019010
[13]

Deyue Zhang, Yukun Guo, Fenglin Sun, Hongyu Liu. Unique determinations in inverse scattering problems with phaseless near-field measurements. Inverse Problems & Imaging, 2020, 14 (3) : 569-582. doi: 10.3934/ipi.2020026
[14]

Suman Kumar Sahoo, Manmohan Vashisth. A partial data inverse problem for the convection-diffusion equation. Inverse Problems & Imaging, 2020, 14 (1) : 53-75. doi: 10.3934/ipi.2019063
[15]

Francis J. Chung. Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem. Inverse Problems & Imaging, 2014, 8 (4) : 959-989. doi: 10.3934/ipi.2014.8.959
[16]

Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems & Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169
[17]

Li Liang. Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data. Inverse Problems & Imaging, 2015, 9 (2) : 469-478. doi: 10.3934/ipi.2015.9.469
[18]

Fioralba Cakoni, Rainer Kress. Integral equations for inverse problems in corrosion detection from partial Cauchy data. Inverse Problems & Imaging, 2007, 1 (2) : 229-245. doi: 10.3934/ipi.2007.1.229
[19]

Soumen Senapati, Manmohan Vashisth. Stability estimate for a partial data inverse problem for the convection-diffusion equation. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021060
[20]

Guangsheng Wei, Hong-Kun Xu. On the missing bound state data of inverse spectral-scattering problems on the half-line. Inverse Problems & Imaging, 2015, 9 (1) : 239-255. doi: 10.3934/ipi.2015.9.239

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (67)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy