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Characterization by approximation
properties
Albert Cohen
This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License †
Abstract The following is a short introduction to Besov spaces and their characterization by means of approx- imation procedures as well as wavelet decompositions. 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 f is function we denote by Pj f its projection onto the space Vj , and by Qj f = Pj+1f − Pj f its projection onto the detail space Wj. The multiscale decomposition of f writes
f = P 0 f +
j≥ 0
Qj f. (1)
The projectors Pj and Qj can be further expressed in terms of biorthogonal scaling functions and wavelets bases:
Pj f :=
|λ|=j
< f, ϕ˜λ > ϕλ and Qj f :=
|λ|=j
< f, ψ˜λ > ψλ. (2)
Here we use the simpli ed notation ϕλ with |λ| = j meaning that the functions are picked at resolution j. In the case where Ω = Rd, these have the general from ϕλ (x) := ϕj,k (x) := 2dj/^2 ϕ
2 j^ x − k
, bur for a general domain Ω = Rd^ proper adaptations of these bases need to be done near the boundary. We can therefore write
f =
dλψλ, dλ :=< f, ψ˜λ >, (3)
where we include in this sum the wavelets at all levels j ≥ 0 and we incorporate the scaling function ϕλ at the rst level |λ| = 0. Under certain assumptions that we shall discuss below, it is known that the Besov norm ‖ f ‖Bsp,q is
equivalent to
‖ P 0 f ‖Lp + ‖
2 sj^ ‖ f − Pj f ‖Lp
j≥ 0 ‖`q^ ,^ (4) ∗Version 1.3: Sep 16, 2009 3:27 pm GMT- †http://creativecommons.org/licenses/by/2.0/
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Connexions module: m19613 2
or to
‖ P 0 f ‖Lp + ‖
2 sj^ ‖ Qj f ‖Lp
j≥ 0 ‖q^.^ (5) Using the equivalence ‖ Qj f ‖Lp ∼ 2 (d/^2 −d/p)j^ ‖ (dλ)|λ|=j ‖p at each level to prove a third equivalent norm in terms of the wavelet coe cients:
‖
2 sj^2 (d/^2 −d/p)j^ ‖ (dλ)|λ|=j ‖`p
j≥ 0
‖`q^. (6)
These equivalences mean that the modulus of smoothness ωn
f, 2 −j^
Lp^ in the de nition of^ B
s p,q can be replaced either by ‖ f − Pj f ‖Lp or by ‖ Qj f ‖Lp. Their validity requires that the spaces Vj satisfy the following two assumptions:
- The Vj must satisfy an approximation property that takes the form of a direct estimate
‖ f − Pj f ‖Lp ≤ Cωn
f, 2 −j^
Lp^.^ (7) Such an estimate ensures that a smooth function will have a fast rate of approximation.
- They must also satisfy smoothness properties that takes the form of an inverse estimate
ωn(fj , t)Lp ≤ C
[
min
1 , t 2 j^
)]n ‖ fj ‖Lp if fj ∈ Vj. (8)
Such an estimate takes into account the smoothness of the spaces Vj : it ensures that a function that is approximated at a su ciently fast rate rate by these spaces should also have some smoothness.
One can show that the direct estimate is satis ed if and only if all polynomials up to order n − 1 can be written as combinations of the scaling functions ϕλ in Vj , or equivalently if the dual wavelets ψ˜λ have n vanishing moments. On the other hand, the inverse estimate requires that the scaling functions ϕλ that generates Vj are smooth in the sense of belonging to W n,p. Note that the direct estimate immediately implies that the expression (4) is less than ‖ f ‖Bp,qs. A more re ned mechanism, using the inverse estimate
(as well as some discrete Hardy inequalities) is used to prove the full equivalence between ‖ f ‖Bsp,q and (5)
or (6). We refer to chapter III in [2] for a detailed proof of these results. These equivalences show that the convergence rate N −t/d^ (N = dim (Vj )) can be achieved by the linear multiscale approximation process f 7 → Pf , if and only if the function has roughly t derivatives in Lp.
References
[1] R. Adams. Sobolev Spaces. Academic Press, 1975.
[2] A. Cohen. Numerical Analysis of Wavelet Methods. Elsevier, 2003.
[3] Ingrid Daubechies. Ten Lectures on Wavelets. SIAM, Philadelphia, PA, 1992. Notes from the 1990 CBMS-NSF Conference on Wavelets and Applications at Lowell, MA.
[4] R. DeVore. Nonlinear Approximation. Acta Numerica, 1998.
[5] B. Jawerth R. DeVore and V. Popov. Compression of wavelet decompositions. American Journal of Math, 114:737285, 1992.
[6] H. Triebel. Theory of Function Spaces. Birkhauser, 1983.
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