= Smooth-mask vanishing-moment criterion
Suppose an integrable <orthonormal> <scaling function> has a <low-pass filter of a multiresolution analysis> that is $C^{p-1}$ near $\pi$, and its <Fourier transform> is $C^{p-1}$ near zero. If the associated <wavelet> has $p$ integrable <vanishing moments>, then $m^{(k)}(\pi)=0$ for $k<p$. Indeed, <moment differentiation of the Fourier transform> makes $\widehat\psi^{(k)}(0)=0$, while $|\widehat\varphi(0)|=1$. In $\widehat\psi(2t)=e^{-it}\overline{m(t+\pi)}\widehat\varphi(t)$, division by the nonzero <smooth> factor proves the conclusion. With only <continuity> of that factor one still obtains a <Peano zero>, but not automatically higher ordinary <derivatives>.
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