I am looking for a transformation in which the gaussian wave when transformed with a particular analysis function would make the energy contents be mostly present only in a short band of frequencies?
Use a dictionary of Gaussian wave atoms, at all different locations and scales (or frequencies). A specific Gaussian should matching maximally with one of the atoms.
This is a generic answer. More details to come if you tell more about your wave: definition, dimension, and purpose of the sparsifying transform