On approximate unsmoothing of functions (Q1979056)

For \(f\in L^1_{\text{loc}} (\mathbb R)\) and \(a>0\) put \[ (T_af)(x)=(2a)^{-1}\int ^a_{-a} f(x+y)dy, \quad x\in \mathbb R. \] The function \(T_af\) and the operator \(T_a\), respectively, is called the smoothing of \(f\) and the smoothing operator. On the other hand, the problem of constructing some \(f\) such that \(T_af=g\), where \(g\) is a given function, is called the unsmoothing problem. Let \(x_i\), \(c_i\in \mathbb R\), \(i=1,\ldots ,n\), and assume that there exists \(f\in L^2(\mathbb R)\) such that \[ (T_af)(x_i) = c_i,\quad i=1,\ldots ,n. \tag{1} \] One cannot hope to recover \(f\) from these data, however, one can try to find some \(\widetilde f\in L^2(\mathbb R)\) such that (1) holds with \(f\) replaced by \(\widetilde f\). This problem is called the approximate unsmoothing problem and the aim of the paper is to give an iterative process which yields its solution.