On the ranges of Mellin multiplier transformations (Q1922981)

A transformation \(T \in [{\mathfrak L}_{\mu, k}]\) such that \((MTf) (s) = m(s) (Mf) (s)\) is called a multiplier transformation in connection with the Mellin transformation \(M\). For the cases in which \(m(s)\) has multiple zeros, theorems concerning the ranges of such multiplier transformations are given here. The results are restated in terms of the bilateral Laplace transformation and hence the Fourier transformation.