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Classical measures of 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.

There exist many different ways of measuring the smoothness of a function ff. The most natural one is certainly the order of differentiability, i.e. the maximal index mm such that f(m)=(ddx)mff(m)=(ddx)mf is continuous. To this particular measure of smoothness, we can associate a class of function spaces: if II is an interval of RR, we denote by Cm(I)Cm(I) the space of continuous functions which have bounded and continuous derivatives, up to the order mm. This space can be equipped with the norm

f C m ( I ) : = sup l = 0 , , m sup x I | f ( l ) ( x ) | . f C m ( I ) : = sup l = 0 , , m sup x I | f ( l ) ( x ) | .
(1)

for which it is a Banach space. (That is, the space is a vector space; the norm satisfies the triangle inequality; f=0f=0 is possible only if f=0f=0; finally, all Cauchy sequences converge: if we have a sequence with entries fnCm(I)fnCm(I) for which fn-fn'fn-fn' can be made arbitrarily small simply by choosing n,n'n,n' sufficiently large, then the fnfn (and all their derivatives up to the mmth) converge uniformly to some function ff in CmCm (and its derivatives).

In the case of a multivariate domain ΩRdΩRd, we define Cm(Ω)Cm(Ω) to be the space of continuous functions which have bounded and continuous partial derivatives αf:=|α|fx1α1xdαdαf:=|α|fx1α1xdαd, for |α|:=α1++αd=0,,m|α|:=α1++αd=0,,m. This space can also be equipped with the norm

f C m ( Ω ) : = | α | m sup x Ω | α f ( x ) | , f C m ( Ω ) : = | α | m sup x Ω | α f ( x ) | ,
(2)

for which it is a Banach space.

In many instances, one is somehow interested in measuring smoothness in an average sense: for this purpose it is natural to introduce the Sobolev spacesWm,p(Ω)Wm,p(Ω) consisting of all functions fLpfLp with partial derivatives up to order mm in LpLp. Here pp is a fixed index in [1,+][1,+]. (Recall that fLp=[Ω|f(x)|p]1pfLp=[Ω|f(x)|p]1p if p<+p<+ and fL=supxΩ|f(x)|fL=supxΩ|f(x)|.) This space is also a Banach space, when equipped with the norm

f W m , p : = | α m α f L p p 1 p . f W m , p : = | α m α f L p p 1 p .
(3)

Note that the norm for CmCm spaces coincides with the Wm,Wm, norm.

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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