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 [3] or [2] for a general treatment): if ff is function we denote by PjfPjf its projection onto the space VjVj, and by Qjf=Pj+1f-PjfQjf=Pj+1f-Pjf its projection onto the detail space WjWj. The multiscale decomposition of ff writes

The projectors PjPj and QjQj can be further expressed in terms of biorthogonal scaling functions and wavelets bases:

Here we use the simplified notation ϕλϕλ with “|λ|=j|λ|=j” meaning that the functions are picked at resolution jj. In the case where Ω=RdΩ=Rd, these have the general from ϕλ(x):=ϕj,k(x):=2dj/2ϕ(2jx-k)ϕλ(x):=ϕj,k(x):=2dj/2ϕ(2jx-k), bur for a general domain Ω=RdΩ=Rd proper adaptations of these bases need to be done near the boundary. We can therefore write

where we include in this sum the wavelets at all levels j≥0j≥0 and we incorporate the scaling function ϕλϕλ at the first level |λ|=0|λ|=0.

Under certain assumptions that we shall discuss below, it is known that the Besov norm ∥f∥Bp,qs∥f∥Bp,qs is equivalent to

or to

Using the equivalence ∥Qjf∥Lp∼2(d/2-d/p)j∥(dλ)|λ|=j∥ℓp∥Qjf∥Lp∼2(d/2-d/p)j∥(dλ)|λ|=j∥ℓp at each level to prove a third equivalent norm in terms of the wavelet coefficients:

These equivalences mean that the modulus of smoothness ωn(f,2-j)Lpωn(f,2-j)Lp in the definition of Bp,qsBp,qs can be replaced either by ∥f-Pjf∥Lp∥f-Pjf∥Lp or by ∥Qjf∥Lp∥Qjf∥Lp. Their validity requires that the spaces VjVj satisfy the following two assumptions:

One can show that the direct estimate is satisfied if and only if all polynomials up to order n-1n-1 can be written as combinations of the scaling functions ϕλϕλ in VjVj, or equivalently if the dual wavelets ψ˜λψ˜λ have nn vanishing moments. On the other hand, the inverse estimate requires that the scaling functions ϕλϕλ that generates VjVj are smooth in the sense of belonging to Wn,pWn,p. Note that the direct estimate immediately implies that the expression Equation 4 is less than ∥f∥Bp,qs∥f∥Bp,qs. A more refined mechanism, using the inverse estimate (as well as some discrete Hardy inequalities) is used to prove the full equivalence between ∥f∥Bp,qs∥f∥Bp,qs and Equation 5 or Equation 6. We refer to chapter III in [2] for a detailed proof of these results.

These equivalences show that the convergence rate N-t/dN-t/d (N= dim (Vj)N= dim (Vj)) can be achieved by the linear multiscale approximation process f↦Pff↦Pf, if and only if the function has roughly “tt derivatives in LpLp”.

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Last edited by Daniel Williamson on Sep 16, 2009 3:27 pm -0500.