(Smoothness of compactly supported wavelets with vanishing moments) Let be an MRA, scaling function and wavelet (see the definition ( Multiresolution analysis ) and propositions ( Scaling equation ),( Existence of orthonormal wavelet bases 2 )). Assume that

1. is compactly supported and (therefore) is finite.

2. , , for some .

Then where and comes from the proposition ( Sufficient conditions for vanishing moments )-c.

The above proposition is to be used with the proposition ( Basic properties of Fourier transform )-7.

the proof may found in [Mallat] , proposition 7.6, p. 287.