A modified Mellin transform is the so-called scale transform of \(x\in L^2(\mathbb{R}_+)\) defined by \[ \widehat x(s)= \int^\infty_0 x(f) f^{2\pi si- 1/2} df\qquad (s\in\mathbb{R}). \] This transform is a natural complement to the Fourier transform for wideband analytic signals. In this paper, limitations for the simultaneous localization of \(x\) and \(\widehat x\) are investigated. Several inequalities are discussed, based on various measures of spread (Heisenberg-type inequalities for variance-like measures, 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. Finally, a new uncertainty relation for the wavelet transform is presented.
For the entire collection see [Zbl 0996.00017].