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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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Localization algorithms for singularities of solutions to convolution equations of the first kind

A. L. Ageev1 / T. V. Antonova2

1Institute of Mathematics and Mechanics UrB RAS, Kovalevskaya st., 16, 620219 Ekaterinburg, Russia. Email:

2Institute of Mathematics and Mechanics UrB RAS, Kovalevskaya st., 16, 620219 Ekaterinburg, Russia. Email:

Citation Information: Journal of Inverse and Ill-posed Problems. Volume 16, Issue 7, Pages 639–650, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: 10.1515/JIIP.2008.039, November 2008

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In this paper we construct and investigate localization algorithms for isolated singularities of a function which is a solution to a linear convolution equation of the first kind whose right-hand side is given with an error. We consider two types of singularities: δ-functions and discontinuities of the first kind. A problem for singularities localization is an ill-posed problem having perturbation. We use averaging methods defined by an averaging functional to obtain the singularities. We formulate conditions that the averaging functional must satisfy. For convolution equations, using the Fourier transform, we obtained upper estimates of precision of the singularities localization, separation threshold and other important characteristics of the supposed methods.

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