Vanishing moments of a wavelet

Summary

A wavelet $\psi$ has $p$ vanishing moments if it is orthogonal to all polynomials of degree less than $p$, that is, $\int x^k \psi(x) \, dx = 0$ for $k = 0, \dots, p - 1$. Then the wavelet coefficients of a function that is smooth near a point are very small at fine scales near that point. This is why wavelets with many vanishing moments compress smooth signals well. The Daubechies wavelets have the shortest support among orthonormal wavelets with a given number of vanishing moments.