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Abstract <p>Integral geometry is one of the most important sections of the theory of ill-posed problems of mathematical physics and analysis. The urgency of the problems of integral geometry is due to the development of tomographic methods, which raise the requirements for the depth of the applied results, the fact that the solution of problems of integral geometry reduces a number of multidimensional inverse problems for partial differential problems, as well as the internal development needs of the theory of ill-posed problems of mathematical physics and analysis. We study the problem of recovering a function from its integrals over conical surfaces in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb{R}^{3}\)</EquationSource> <!--LobJMat2561127Akrambegmatov-m1--> </InlineEquation>, where the integration involves a singular weight. We establish the connection between the structure of the weight function and the degree of ill-posedness of the inverse problem. Stability estimates are obtained for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\beta&lt;1\)</EquationSource> <!--LobJMat2561127Akrambegmatov-m2--> </InlineEquation>, and strong ill-posedness is proved for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\beta\geq 1\)</EquationSource> <!--LobJMat2561127Akrambegmatov-m3--> </InlineEquation>. We propose regularization methods, present generalizations with weighted transforms, and formulate.</p>

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On Stability and Ill-Posedness in the Inverse Problem of Integral Geometry over Conical Surfaces with Singular Weights

  • 期刊论文

  • 发表日期：

    2025年11月1日

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  • Akram Begmatov,
  • Alisher Ismoilov

摘要

Abstract

Integral geometry is one of the most important sections of the theory of ill-posed problems of mathematical physics and analysis. The urgency of the problems of integral geometry is due to the development of tomographic methods, which raise the requirements for the depth of the applied results, the fact that the solution of problems of integral geometry reduces a number of multidimensional inverse problems for partial differential problems, as well as the internal development needs of the theory of ill-posed problems of mathematical physics and analysis. We study the problem of recovering a function from its integrals over conical surfaces in \(\mathbb{R}^{3}\) , where the integration involves a singular weight. We establish the connection between the structure of the weight function and the degree of ill-posedness of the inverse problem. Stability estimates are obtained for \(\beta<1\) , and strong ill-posedness is proved for \(\beta\geq 1\) . We propose regularization methods, present generalizations with weighted transforms, and formulate.