TITLE: What is a "vanishing moment"?
QUESTION [25 upvotes]: In this paper, Sweldens says about desireable properties of wavelets:

To analyze and represent such signals we need wavelets that are local in space and frequency.  Typically this is achieved by building wavelets which have compact support (localization in space), which are smooth (decay towards high frequencies), and which have _vanishing moments_ (decay towards low frequencies).

I understand that compact support means the function is non-zero on a limited portion of its domain.  "Smoothness" I believe means that there are no corners in the basis function, ie it would not take infinitely high frequencies to represent the curve (using Fourier analysis).
I do not know what a "vanishing moment" means, intuitively, though. Why would you call a "decay to towards low frequencies" a vanishing moment?

REPLY [5 votes]: The n-th moment of a function is equal to the n-th derivative of its Fourier transform at zero frequency. So higher the number of zero moments, higher the number of zero derivatives and smoother the signal decays from mid frequency to DC in the frequency domain.