Vanishing moment ($\int x^r\psi (x)dx=0$)

A wavelet has $q$ vanishing moments when these integrals vanish for $0\le r<q$. For compact support, this is equivalent to a zero of order at least $q$ at the origin of its Fourier transform. Such a wavelet annihilates polynomials of degree less than $q$, and a Taylor polynomial bounds coefficients on smooth regions.