Vanishing moments of a wavelet

Remarks

Equivalently, $\int_\R P(x) \psi(x) \, dx = 0$ for every polynomial $P$ of degree less than $p$. In terms of the Fourier transform, under the same integrability, this says that $\hat\psi$ has a zero of order at least $p$ at $\xi = 0$. The number of vanishing moments of $\psi$ is the largest such $p$.

Example: the Haar wavelet $\psi = 1_{[0, 1/2)} - 1_{[1/2, 1)}$ has $\int \psi = \tfrac12 - \tfrac12 = 0$, so it has $1$ vanishing moment.

Non-example: the Haar wavelet does not have $2$ vanishing moments, since $\int_\R x \psi(x) \, dx = \tfrac18 - \tfrac38 = -\tfrac14 \ne 0$.