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

A note on analyticity properties of far field patterns

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.
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. doi: 10.4153/CJM-1961-054-1.

[2]

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

[3]

D. L. Colton and R. Kress, "Integral Equation Methods in Scattering Theory,", John Wiley & Sons, (1983).

[4]

D. L. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", $2^{nd}$ edition, (1998).

[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. doi: 10.1007/BF01448415.

[6]

L. Hörmander, "An Introduction to Complex Analysis in Several Variables,", $3^{rd}$ edition, (1990).

[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. doi: 10.1137/120872164.

[8]

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

[9]

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

[10]

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

[11]

S. G. Krantz, "Function Theory of Several Complex Variables,", $2^{nd}$ edition, (1992).

[12]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications,", I, I (1972).

[13]

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

[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. doi: 10.1007/BF01077418.

[15]

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

show all references

References:
[1]

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

[2]

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

[3]

D. L. Colton and R. Kress, "Integral Equation Methods in Scattering Theory,", John Wiley & Sons, (1983).

[4]

D. L. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", $2^{nd}$ edition, (1998).

[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. doi: 10.1007/BF01448415.

[6]

L. Hörmander, "An Introduction to Complex Analysis in Several Variables,", $3^{rd}$ edition, (1990).

[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. doi: 10.1137/120872164.

[8]

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

[9]

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

[10]

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

[11]

S. G. Krantz, "Function Theory of Several Complex Variables,", $2^{nd}$ edition, (1992).

[12]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications,", I, I (1972).

[13]

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

[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. doi: 10.1007/BF01077418.

[15]

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

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