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Collection by: Albert Cohen. E-mail the author

# Besov spaces and nonlinear approximation

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.

A natural idea for approximating a function ff


by wavelets is to retain in (Reference) the NN largest contributions in the norm in which we plan to measure the error. In the case where this norm is LpLp, this is given by

A N f : = λ E N , p ( f ) d λ ψ λ , A N f : = λ E N , p ( f ) d λ ψ λ ,
(1)

where EN,p(f)EN,p(f) is the set of indices of the NN largest dλψλLpdλψλLp. This set depends on the function ff, making this approximation process nonlinear. Other instances of nonlinear approximation are discussed in [4].

An important result established in [5] states that f-ANfLpN-r/df-ANfLpN-r/d is achieved for functions fBq,qrfBq,qr where 1/q=1/p+r/d1/q=1/p+r/d. Note that this relation between pp and qq corresponds to a critical case of the Sobolev embedding of Bq,qrBq,qr into LpLp. In particular, Bq,qrBq,qr is not contained in Bp,pεBp,pε for any ε>0ε>0, so that no decay rate can be achieved by a linear approximation process for all the functions ff in the space Bq,qrBq,qr. (For some functions in Bq,qrBq,qr, which happen to also lie in spaces for which an independent linear approximation theorem can be written, it is of course possible to get a linear approximation rate; the point here is that this is possible only via such additional information.)

Note also that for large values of rr, the parameter qq given by 1/q=1/p+r/d1/q=1/p+r/d is smaller than 1. In such a situation the space Bq,qsBq,qs is not a Banach space any more and is only a quasi-norm (it fails to satisfy the triangle inequality x+yx+yx+yx+y). However, this space is still contained in L1L1 (by a Sobolev-type embedding) and its characterization by means of wavelets coefficients according to still holds. Letting qq go to zero as rr goes to infinity allows the presence of singularities in the functions of Bq,qrBq,qr even when rr is large: for example, a function which is piecewise CnCn on an interval except at a finite number of isolated points of discontinuities belongs to all Bq,qrBq,qr for q<1/sq<1/s and r<nr<n. This is a particular instance where a non-linear approximation process will perform substantially better than a linear projection.

## References

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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