A fast decaying wavelet has n vanishing moments if and only if its is the n th derivative of a fast decaying function. If this error can be uniformly estimated on an interval, this yields a tool for regularity analysis on an interval. This condition relates the pointwise regularity of a signal to the decay of its wavelet transform's modulus. It can be extended to an interval and, of course, to the whole real axis. A signal and its wavelet transform, computed with the derivative of a Gaussian.
Finer scales are at top.
Zero coefficients are represented by a medium gray. Hence, the regular parts are medium gray. Notice the cones below the singularities.