On the reconstruction of convolution-type operators from inaccurate information

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.