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An important feature of Besov spaces is that they admit equivalent characterization by multiresolution approximation properties and by wavelet decompositions.
Here we use the following standard notation (see [link] or [link] for a general treatment): if is function we denote by its projection onto the space , and by its projection onto the detail space . The multiscale decomposition of writes
The projectors and can be further expressed in terms of biorthogonal scaling functions and wavelets bases:
Here we use the simplified notation with “ ” meaning that the functions are picked at resolution . In the case where , thesehave the general from , bur for a general domain proper adaptations of these bases need to be done near the boundary. We can thereforewrite
where we include in this sum the wavelets at all levels and we incorporate the scaling function at the first level .
Under certain assumptions that we shall discuss below, it is known that the Besov norm is equivalent to
or to
Using the equivalence at each level to prove a third equivalent norm interms of the wavelet coefficients:
These equivalences mean that the modulus of smoothness in the definition of can be replaced either by or by . Their validity requires thatthe spaces satisfy the following two assumptions:
One can show that the direct estimate is satisfied if and only if all polynomials up to order can be written as combinations of the scaling functions in , or equivalently if the dual wavelets have vanishing moments. On the other hand, the inverse estimate requires that the scaling functions that generates are smooth in the sense of belonging to . Note that the direct estimate immediately implies that theexpression [link] is less than . A more refined mechanism, using theinverse estimate (as well as some discrete Hardy inequalities) is used to prove the full equivalence between and [link] or [link] . We refer to chapter III in [link] for a detailed proof of these results.
These equivalences show that the convergence rate ( ) can be achieved by the linearmultiscale approximation process , if and only if the function has roughly “ derivatives in ”.
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