(Vanishing moments vs decay at infinity) Suppose that and the collection is -orthogonal with respect to the scale index:

For any if and then , .

We provide a proof by induction in .

In the case we have for : We use the proposition ( Basic properties of Fourier transform )-3. We use the formula ( Property of scale and transport 6 ). Hence, We set and choose so that for some to be determined later. By the inclusion and the proposition ( Basic properties of Fourier transform )-5 By the inclusion and the proposition ( Dominated convergence theorem ), We choose so that then

For general , we assume that the statement is already proven for . We have By inclusion we have thus We continue The integral in the above estimate for is bounded as by inclusion . By induction assumption , . Hence, we cancel and pass to obtain