On inverse problem leading to second-order linear functionals (Q968968)

The main result of this paper is the fact that, for every real number \(\epsilon\) and every positive-definite linear functional \(L\), the (unique) linear functional \(L_\epsilon\), satisfying the distributional equation \(L_\epsilon-\epsilon L'_{\epsilon}=L\), is also positive-definite. Then, a necessary and sufficient condition for \(L_{\epsilon}\) to be semiclassical is proved. Finally, for the case when \(L\) is the Laguerre linear functional with parameter \(\alpha=0\), an integral representation of \(L_{\epsilon}\) is given.