(Vanishing moments vs approximation) Assume that

1. a function ,

2. the derivative is bounded on : for some ,

3. a function has compact support,

4. , ,

5.

then there exists a constant such that

We assume without loss of generality that see the notation of the section ( Elementary definitions of wavelet analysis ). We estimate directly: We use the formula ( Property of scale and transport 1 ) and the proposition ( Taylor decomposition in Schlomilch, Lagrange and Cauchy forms ). Note that is the middle point of . where . We use the condition (4). We use the condition (2). We use the formula ( Holder inequality ).