Berger measure for some transformations of subnormal weighted shifts

A subnormal weighted shift may be transformed to another shift in various ways, such as taking the p-th power of each weight or forming the Aluthge transform. \ We determine in a number of cases whether the resulting shift is subnormal, and, if it is, find a concrete representation of the associated Berger measure, directly for finitely atomic measures, and using both Laplace transform and Fourier transform methods for more complicated measures. \ Alternatively, the problem may be viewed in purely measure-theoretic terms as the attempt to solve moment matching equations such as $(\int t^n \, d\mu(t))^2 = \int t^n \, d\nu(t)$ ($n=0, 1, \ldots$) for one measure given the other.