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This article provides a framework to regularize operator equations of the first kindwhere the underlying operator is linear and continuous between distribution spaces, the dual spacesof smooth functions. To regularize such a problem, the authors extend Louis’ method of approximateinverse from Hilbert spaces to distribution spaces. The idea is to approximate the exact solution inthe weak topology ... read moreby a smooth function, where the smooth function is generated by a mollifier. Theresulting regularization scheme consists of the evaluation of the given data at so-called reconstructionkernels which solve the dual operator equation with the mollifier as right-hand side. A nontrivialexample of such an operator is given by the spherical Radon transform which maps a function toits mean values over spheres centered on a line or plane. This transform is one of the mathematicalmodels in sonar and radar. After establishing the theory of the approximate inverse for distributions,we apply it to the spherical Radon transform. The article also contains numerical results.read less
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