Wavelet transform regularity

The regularity condition requires that the mother wavelet has to be locally smooth and concentrated in both the time and frequency domains. To ascertain a good local property in the frequency domain, the W f a,b) is required to decay rapidly with decrease of scale a. As a result this requires the first m moments of x) equal to zero. These are the vanishing moments. If the Fourier transform of the wavelet is M times differentiable the wavelet has M vanishing moments. 

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