Geodesic
On the reconstruction of convolution-type operators from inaccurate information
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and differential equations, Tome 269 (2010), pp. 181-192.

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We address the problem of optimal reconstruction of the values of a linear operator on $\mathbb R^d$ or $\mathbb Z^d$ from approximate values of other operators. Each operator acts as the multiplication of the Fourier transform by a certain function. As an application, we present explicit expressions for optimal methods of reconstructing the solution of the heat equation (for continuous and difference models) at a given instant of time from inaccurate measurements of this solution at other time instants. 
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G. G. Magaril-Il'yaev; K. Yu. Osipenko. On the reconstruction of convolution-type operators from inaccurate information. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and differential equations, Tome 269 (2010), pp. 181-192. http://geodesic.mathdoc.fr/item/TM_2010_269_a14/

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