Inequalities in Mellin-Fourier signal analysis

A specific form of the Mellin transform, referred to as the “scale transform,” is known to be a natural complement to the Fourier transform for wideband analytic signals. In this paper, limitations for the simultaneous localization of scale transform pairs are investigated. A number of inequalities are established and discussed, based on various measures of spread (Heisenberg-type inequalities for variance-like measures and Hirschman-type inequalities for entropy). The same issue of maximally concentrating a signal in both scale and frequency domains is also addressed via spread measures which are applied directly to joint scale-frequency distributions. A simple way of obtaining inequalities for Altes-type distributions is pointed out, new results pertaining to the unitary Bertrand distribution are established, as well as a new form of uncertainty relation for the wavelet transform.