Reconstruction of continuous functions from locally uniform weighted averages (Q408298)

Consider the equation NEWLINE\[NEWLINE f \ast \mu = gNEWLINE\]NEWLINE on the real line. It is proved that, under certain assumptions on the measure \(\mu,\) the above equation can be solved if \(g \in C^r (\mathbb R).\) The main theorem states that, if \(\mu\) is a compactly supported absolutely continuous measure whose density is given by a finite linear combination of indicator functions, then for \(g \in C^{r+1}(\mathbb R)\) there exists \(f \in C^r(\mathbb R)\) such that NEWLINE\[NEWLINEf \ast \mu = g.NEWLINE\]NEWLINE First, this is established with additional assumptions on the measure \(\mu\) and the function \(g.\) Here is a sample: Let \(g \in C^{r+1}(\mathbb R)\) and \( \mu,\) a compactly supported measure which is absolutely continuous with respect to the Lebesgue measure on \(\mathbb R.\) Assume that the density of \(\mu\) is given by \(c_1 \chi_{[a_1, b_1]} + c_2 \chi_{[a_2, b_2]}\) where \([a_1, b_1] \cap [a_2, b_2] \) is empty or a singleton. If the support of \(g\) is contained in \((-\infty, \beta)\) or in \((\beta, \infty),\) for some \(\beta \in \mathbb R,\) then there exists \(f \in C^r(\mathbb R)\) such that \(f \ast \mu = g.\) The author's methods explicitly construct \(f\) from \(g\) and \(\mu.\)