Scale limitedness and sampling rate of time limited signals

A signal is called scale-limited if the wavelet transform is zero for all scales below a fixed value and for all shifts. The practical considerations due to which this concept has emerged, have been discussed. As an extension the δ₀ practical scale-limitedness is defined. A signal is called δ₀-practically scale-limited to a_l if the fraction of the energy for the scale values less than a_l and for all shifts is less than δ₀. Finally it is proved that under reasonable assumptions, for the same error bound on the reconstruction of time-limited signals, the signals with smaller scale-limits should be sampled at a higher rate