Results 1  10
of
14
Inverse Scattering On Asymptotically Hyperbolic Manifolds
 ACTA MATH
, 1998
"... Scattering is defined on compact manifolds with boundary which are equipped with an asymptotically hyperbolic metric, g: A model form is established for such metrics close to the boundary. It is shown ..."
Abstract

Cited by 27 (2 self)
 Add to MetaCart
Scattering is defined on compact manifolds with boundary which are equipped with an asymptotically hyperbolic metric, g: A model form is established for such metrics close to the boundary. It is shown
Explicit inversion formulae for the spherical mean radon transform
 Inverse Problems
"... We derive explicit formulas for the reconstruction of a function from its integrals over a family of spheres, or for the inversion of the spherical mean Radon transform. Such formulas are important for problems of thermo and photo acoustic tomography. A closedform inversion formula of a filtratio ..."
Abstract

Cited by 20 (7 self)
 Add to MetaCart
We derive explicit formulas for the reconstruction of a function from its integrals over a family of spheres, or for the inversion of the spherical mean Radon transform. Such formulas are important for problems of thermo and photo acoustic tomography. A closedform inversion formula of a filtrationbackprojection type is found for the case when the centers of the integration spheres lie on a sphere in R n surrounding the support of the unknown function. An explicit series solution is presented for the case when the centers of the integration spheres lie on a general closed surface.
Range descriptions for the spherical mean Radon transform, preprint 2006, arXiv: math. AP/0606314
"... The transform considered in the paper averages a function supported in a ball in R n over all spheres centered at the boundary of the ball. This Radon type transform arises in several contemporary applications, e.g. in thermoacoustic tomography and sonar and radar imaging. Range descriptions for suc ..."
Abstract

Cited by 17 (10 self)
 Add to MetaCart
The transform considered in the paper averages a function supported in a ball in R n over all spheres centered at the boundary of the ball. This Radon type transform arises in several contemporary applications, e.g. in thermoacoustic tomography and sonar and radar imaging. Range descriptions for such transforms are important in all these areas, for instance when dealing with incomplete data, error correction, and other issues. Four different types of complete range descriptions are provided, some of which also suggest inversion procedures. Necessity of three of these (appropriately formulated) conditions holds also in general domains, while the complete discussion of the case of general domains will require another publication.
Integral representations and Liouville theorems for solutions of periodic elliptic equations
 J. Funct. Anal
"... The paper contains integral representations for certain classes of exponentially growing solutions of second order periodic elliptic equations. These representations are the analogs of those previously obtained by S. Agmon, S. Helgason, and other authors for solutions of the Helmholtz equation. When ..."
Abstract

Cited by 9 (6 self)
 Add to MetaCart
The paper contains integral representations for certain classes of exponentially growing solutions of second order periodic elliptic equations. These representations are the analogs of those previously obtained by S. Agmon, S. Helgason, and other authors for solutions of the Helmholtz equation. When one restricts the class of solutions further, requiring their growth to be polynomial, one arrives to Liouville type theorems, which describe the structure and dimension of the spaces of such solutions. The Liouville type theorems previously proved by M. Avellaneda and F.H. Lin, and J. Moser and M. Struwe for periodic second order elliptic equations in divergence form are significantly extended. Relations of these theorems with the analytic structure of the Fermi and Bloch surfaces are explained.
Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds
, 2004
"... ..."
The Helmholtz equation on Lipschitz domains
, 1995
"... We use the method of layer potentials to study interior and exterior Dirichlet and Neumann problems for the Helmholtz equation ( + k2)u = 0 on a Lipschitz domain for all wave number k 2C with Imk 0. Following the approach for the case of smooth boundary [3], we pursue as solution a single layer pote ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
We use the method of layer potentials to study interior and exterior Dirichlet and Neumann problems for the Helmholtz equation ( + k2)u = 0 on a Lipschitz domain for all wave number k 2C with Imk 0. Following the approach for the case of smooth boundary [3], we pursue as solution a single layer potential for Neumann problem or a double layer potential for Dirichlet problem. The lack of smoothness of a Lipschitz boundary brought some additional di culties. These are overcome through the use of harmonic analysis techniques together with a careful study of the properties of layer potentials near the boundary and the spectra of the traces of the layer potentials.
A family of inversion formulas in thermoacoustic tomography
, 2009
"... We present a family of closed form inversion formulas in thermoacoustic tomography in the case of a constant sound speed. The formulas are presented in both timedomain and frequencydomain versions. As special cases, they include most of previously known filtered backprojection type formulas. 1 Int ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We present a family of closed form inversion formulas in thermoacoustic tomography in the case of a constant sound speed. The formulas are presented in both timedomain and frequencydomain versions. As special cases, they include most of previously known filtered backprojection type formulas. 1 Introduction and statement of main results Thermoacoustic tomography has recently attracted considerable attention as a promising method of biomedical imaging [16, 20, 21, 24, 28]. We give here a brief introduction to the mathematical model of TAT. An electromagnetic (EM) pulse in visible light or radiofrequency range
unknown title
, 2006
"... Range descriptions for the spherical mean Radon transform. I. Functions supported in a ball. ∗ ..."
Abstract
 Add to MetaCart
Range descriptions for the spherical mean Radon transform. I. Functions supported in a ball. ∗
INTEGRAL REPRESENTATIONS OF SOLUTIONS OF PERIODIC ELLIPTIC EQUATIONS
, 2006
"... Dedicated to Stas Molchanov on the occasion of his 65th birthday Abstract. The paper discusses relations between the structure of the complex Fermi surface below the spectrum of a second order periodic elliptic equation and integral representations of certain classes of its solutions. These integral ..."
Abstract
 Add to MetaCart
Dedicated to Stas Molchanov on the occasion of his 65th birthday Abstract. The paper discusses relations between the structure of the complex Fermi surface below the spectrum of a second order periodic elliptic equation and integral representations of certain classes of its solutions. These integral representations are analogs of those previously obtained by S. Agmon, S. Helgason, and other authors for solutions of the Helmholtz equation (i.e., for generalized eigenfunctions of Laplace operator). In a previous joint work with Y. Pinchover we described all solutions that can be represented as integrals of positive Bloch solutions over the imaginary Fermi surface, with a hyperfunction as a “measure”. Here we characterize the class of solutions such that the corresponding hyperfunction is a distribution on the Fermi surface. 1.