0.5 Besov spaces and nonlinear approximation

A natural idea for approximating a function $f$ by wavelets is to retain in [link] the $N$ largest contributions in the norm in which we plan to measure the error. In the case wherethis norm is $L^p$ , this is given by

where $E_{N,p}\left(f\right)$ is the set of indices of the $N$ largest $\parallel d_{\lambda }{\psi }_{\lambda }{\parallel }_{L^p}$ . This set depends on the function $f$ , making this approximation process nonlinear . Other instances of nonlinear approximation are discussed in [link] .

An important result established in [link] states that $\parallel f-A_N{f\parallel }_{L^p}\sim N^{-r/d}$ is achieved for functions $f\in B_{q,q}^r$ where $1/q=1/p+r/d$ . Note that this relation between $p$ and $q$ corresponds to a critical case of the Sobolev embedding of $B_{q,q}^r$ into $L^p$ . In particular, $B_{q,q}^r$ is not contained in $B_{p,p}^{\epsilon }$ for any $\epsilon >0$ , so that no decay rate can be achieved by a linearapproximation process for all the functions $f$ in the space $B_{q,q}^r$ . (For some functions in $B_{q,q}^r$ , which happen to also lie in spaces for which an independent linear approximation theorem can be written, it isof course possible to get a linear approximation rate; the point here is that this ispossible only via such additional information.)

Note also that for large values of $r$ , the parameter $q$ given by $1/q=1/p+r/d$ is smaller than 1. In such a situation the space $B_{q,q}^s$ is not a Banach space any more and is only a quasi-norm (it fails to satisfythe triangle inequality $\parallel x+y\parallel \le \parallel x\parallel +\parallel y\parallel$ ). However, this space is still contained in $L^1$ (by a Sobolev-type embedding) and its characterization by means of wavelets coefficients according tostill holds. Letting $q$ go to zero as $r$ goes to infinity allows the presence of singularities in the functions of $B_{q,q}^r$ even when $r$ is large: for example, a function which is piecewise ${\mathcal{C}}^n$ on an interval except at a finite number of isolated points of discontinuities belongsto all $B_{q,q}^r$ for $q<1/s$ and $r<n$ . This is a particular instance where a non-linear approximation process will performsubstantially better than a linear projection.