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    "This is an introduction to fractional measures of smoothness for functions via Besov spaces that generalize the classical Sobolev smoothness spaces."

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Towards fractional smoothness

Module by: Albert Cohen. E-mail the author

Summary: The following is a short introduction to Besov spaces and their characterization by means of approximation procedures as well as wavelet decompositions.

All the above spaces share the common feature that the regularity index is an integer. In many applications, one is interested to allow fractional order of smoothness, in order to describe the regularity of a function in a more precise way. The question thus arises of how to fill the gaps between integer smoothness classes. There are at least two instances where such a generalization is very natural:

  • In the case of L2L2-Sobolev spaces Hm:=Wm,2Hm:=Wm,2 and when Ω=RdΩ=Rd, we can define an equivalent norm based on the Fourier transform, since by Parseval's formula we have the norm equivalence
    fHm2Rd(1+|ω|)2m|f^(ω)|2dω.fHm2Rd(1+|ω|)2m|f^(ω)|2dω.
    (1)
    For a non-integer s0s0, it is thus natural to define the space HsHs as the set of all L2L2 functions such that
    fHs2:=Rd(1+|ω|)2s|f^(ω)|2dω,fHs2:=Rd(1+|ω|)2s|f^(ω)|2dω,
    (2)
    is finite.
  • In the case of CmCm spaces, we note that supxΩ|f(x)-f(x-h)|C|h|supxΩ|f(x)-f(x-h)|C|h| if fC1fC1 for any hRdhRd whereas for an arbitrary function fC0fC0, supxΩ|f(x)-f(x-h)|supxΩ|f(x)-f(x-h)| might go to zero arbitrarily slow as |h|0|h|0. This motivates the definition of the Hölder spaceCsCs, 0<s<10<s<1 consisting of those fC0fC0 such that
    supxΩ|f(x)-f(x-h)|C|h|s.supxΩ|f(x)-f(x-h)|C|h|s.
    (3)
    If m<s<m+1m<s<m+1, a natural definition of CsCs is given by fCmfCm and αfCs-mαfCs-m, |α|=m|α|=m. It can be proved that this property can also be expressed by
    supxΩ|Δhnf(x)|C|h|s,supxΩ|Δhnf(x)|C|h|s,
    (4)
    where n>sn>s and ΔhnΔhn is the nn-th order finite difference operator defined recursively by Δh1f(x)=f(x)-f(x-h)Δh1f(x)=f(x)-f(x-h) and Δhnf(x)=Δh1(Δhn-1)f(x)Δhnf(x)=Δh1(Δhn-1)f(x) (for example Δh2f(x)=f(x)-2f(x-h)+f(x-2h)Δh2f(x)=f(x)-2f(x-h)+f(x-2h)). When ss is not an integer, the spaces CsCs that we have defined are also denoted as Ws,Ws,. The space CsCs can be equiped with the norm
    fCs(Ω):=fL(Ω)+suphRd|h|-sΔhnfL(Ω).fCs(Ω):=fL(Ω)+suphRd|h|-sΔhnfL(Ω).
    (5)

Let us give two important instances in which the above spaces appear in a natural way. The first is the study of the restriction of a function f(x1,,xd)f(x1,,xd) to a manifold of lower dimension, for example the hyperplane defined by xd=0xd=0. If g(x1,,xd-1)=f(x1,,xd-1,0)g(x1,,xd-1)=f(x1,,xd-1,0) is such a restriction, then it is known that fHs(RdfHs(Rd for s>12s>12 implies that gHs-12(Rd-1)gHs-12(Rd-1). The second one is the study of the Brownian motion W(t)W(t) on an interval II, for which it is known that W(t)W(t) is almost surely in C12-εC12-ε for all ε>0ε>0.

References

  1. Adams, R. (1975). Sobolev Spaces. Academic Press.
  2. Cohen, A. (2003). Numerical Analysis of Wavelet Methods. Elsevier.
  3. Daubechies, Ingrid. (1992). Ten Lectures on Wavelets. [Notes from the 1990 CBMS-NSF Conference on Wavelets and Applications at Lowell, MA]. Philadelphia, PA: SIAM.
  4. DeVore, R. (1998). Nonlinear Approximation. Acta Numerica.
  5. R. DeVore, B. Jawerth, and V. Popov,. (1992). Compression of wavelet decompositions. American Journal of Math, 114, 737-285.
  6. H. Triebel. (1983). Theory of Function Spaces. Birkhauser.

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