A classic application of the Mellin transform is in radar engineering. Recall that the resolution in time of a signal is the reciprocal of the spread of its Fourier transform. In radar (or sonar) a signal is reflected from a moving target and you would like to accurately determine its velocity. The velocity resolution of the signal is the reciprocal of the spread of its Mellin transform.

<A HREF="http://www.norbertwiener.umd.edu/crowds/documents/Ovarlez93.pdf">Cramer Rao bound computation for velocity estimation in the broad-band case using the Mellin transform</A>