On approximate unsmoothing of functions

Abstract

The smoothing \(T_af\), for \(a \gt 0\), of a locally-integrable function \(f : \mathbb{R} \rightarrow \mathbb{R}\) is defined by \[(T_af)(x) := \frac{1}{2a} \int^{+a}_{-a} f(x+y) dy, \quad x \in \mathbb{R}.\] For a given \(g : \mathbb{R} \rightarrow \mathbb{R}\), any solution \(f\) of the equation \(T_af = g\) is called an unsmoothing of \(g\). In this note we analyse the problem of constructing a function \(\tilde{f} : \mathbb{R} \rightarrow \mathbb{R}\) such that \((T_a \tilde{f})(x_i) = g(x_i)\) for a given set of points \(x_1,x_2, \ldots , x_n \in \mathbb{R}\). We give an iterative process of constructing such an \(\tilde{f}\) under the assumption \(f \in L_2(\mathbb{R})\).