On the condition number of the antireflective transform |
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Authors: | Marco Donatelli Martin Hanke |
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Institution: | a Dipartimento di Fisica e Matematica, Università dell’Insubria, Via Valleggio 11, 22100 Como, Italy b Institut für Mathematik, Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany |
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Abstract: | Deconvolution problems with a finite observation window require appropriate models of the unknown signal in order to guarantee uniqueness of the solution. For this purpose it has recently been suggested to impose some kind of antireflectivity of the signal. With this constraint, the deconvolution problem can be solved with an appropriate modification of the fast sine transform, provided that the convolution kernel is symmetric. The corresponding transformation is called the antireflective transform. In this work we determine the condition number of the antireflective transform to first order, and use this to show that the so-called reblurring variant of Tikhonov regularization for deconvolution problems is a regularization method. Moreover, we establish upper bounds for the regularization error of the reblurring strategy that hold uniformly with respect to the size n of the algebraic system, even though the condition number of the antireflective transform grows with n. We briefly sketch how our results extend to higher space dimensions. |
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Keywords: | 65F22 65R32 65T50 |
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