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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?

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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

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Gabor transform and Morlet Wavelet could help you. Morlet Wavelet scaling functions are gaussians, whereas the imaginary and real parts of the wavelet can be tuned in frequency to be first and second differential approximators of gaussians.

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