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 ∥dλψλ∥Lp∥dλψλ∥Lp.
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-ANf∥Lp∼N-r/d∥f-ANf∥Lp∼N-r/d is achieved for functions f∈Bq,qrf∈Bq,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+y∥≤∥x∥+∥y∥∥x+y∥≤∥x∥+∥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.
-
Adams, R. (1975). Sobolev Spaces. Academic Press.
-
Cohen, A. (2003). Numerical Analysis of Wavelet Methods. Elsevier.
-
Daubechies, Ingrid. (1992). Ten Lectures on Wavelets. [Notes from the 1990 CBMS-NSF Conference on Wavelets and Applications at Lowell, MA]. Philadelphia, PA: SIAM.
-
DeVore, R. (1998). Nonlinear Approximation. Acta Numerica.
-
R. DeVore, B. Jawerth, and V. Popov,. (1992). Compression of wavelet decompositions. American Journal of Math, 114, 737-285.
-
H. Triebel. (1983). Theory of Function Spaces. Birkhauser.
