The definition of “order of smoothness ss in LpLp” for ss non-integer and
pp different from 2 or ∞∞ is more subject to arbitrary choices.
Among others, one may consider:
- Sobolev spaces Ws,pWs,p defined (if m<s<m+1m<s<m+1) by
∥f∥Ws,p:=∥f∥Wm,pp+∑|α|=m∫Ω×Ω|∂αf(x)-∂αf(y)|p|x-y|(s-m)p+ddxdy1p∥f∥Ws,p:=∥f∥Wm,pp+∑|α|=m∫Ω×Ω|∂αf(x)-∂αf(y)|p|x-y|(s-m)p+ddxdy1p
(1) - Bessel-potential spaces Hs,pHs,p defined by means of the Fourier transform
operator FF,
∥f∥Hs,p=∥f∥Lpp+∥F-1(1+|·|s)Ff∥Lpp1p.∥f∥Hs,p=∥f∥Lpp+∥F-1(1+|·|s)Ff∥Lpp1p.
(2) - Besov spaces Bp,qsBp,qs, involving an extra parameter qq
that we define below through finite differences. These spaces include most
of those
that we have listed so far as particular cases. As we shall see,
one of their main interest is that they can be exactly
characterized by multiresolution approximation error,
as well as from the size properties of the wavelet coefficients.
We define the nn-th order LpLp modulus of smoothness of ff by
ω
n
(
f
,
t
)
L
p
=
sup
|
h
|
≤
t
∥
Δ
h
n
f
∥
L
p
(
Ω
h
,
n
)
,
ω
n
(
f
,
t
)
L
p
=
sup
|
h
|
≤
t
∥
Δ
h
n
f
∥
L
p
(
Ω
h
,
n
)
,
(3)where Ωh,n:={x∈Ω;x-kh∈Ω,k=0,⋯,n}Ωh,n:={x∈Ω;x-kh∈Ω,k=0,⋯,n}.
Here we measure the “size” of ΔhnfΔhnf in
LpLp-norm,
where we restrict to Lp(Ωh,n)Lp(Ωh,n) to ensure that all the arguments
x-khx-kh
occurring in the computation of Δhnf(x)Δhnf(x) still live in ΩΩ.
For p,q≥1p,q≥1, s>0s>0, the Besov spaces Bp,qsBp,qs consists of those
functions f∈Lpf∈Lp such that
(
2
s
j
ω
n
(
f
,
2
-
j
)
L
p
)
j
≥
0
∈
ℓ
q
.
(
2
s
j
ω
n
(
f
,
2
-
j
)
L
p
)
j
≥
0
∈
ℓ
q
.
(4)Here nn is an integer strictly larger than ss.
A natural norm for such a space is then given by
∥
f
∥
B
p
,
q
s
:
=
∥
f
∥
L
p
+
∥
(
2
s
j
ω
n
(
f
,
2
-
j
)
L
p
)
j
≥
0
∥
ℓ
q
.
∥
f
∥
B
p
,
q
s
:
=
∥
f
∥
L
p
+
∥
(
2
s
j
ω
n
(
f
,
2
-
j
)
L
p
)
j
≥
0
∥
ℓ
q
.
(5)If q=∞q=∞, the condition Equation 4 simply means that
∥Δhnf∥Lp≤Ch-s∥Δhnf∥Lp≤Ch-s for |h|≤1|h|≤1. For q<∞q<∞, the decay condition on ΔhnfΔhnf
is slightly stronger, since we require that the sequence
(2sjωn(f,2-j)Lp)j≥0qi(2sjωn(f,2-j)Lp)j≥0qi be summable.
The space Bp,qsBp,qs also represents “ss order of smoothness measured in
LpLp";
the parameter qq allows a finer tuning on
the degree of smoothness - one has Bp,q1s⊂Bp,q2sBp,q1s⊂Bp,q2s
if q1≤q2q1≤q2 - but plays a minor role in comparison to ss since
clearly
B
p
,
q
1
s
1
⊂
B
p
,
q
2
s
2
,
if
s
1
≥
s
2
,
B
p
,
q
1
s
1
⊂
B
p
,
q
2
s
2
,
if
s
1
≥
s
2
,
(6)regardless of the values of q1q1 and q2q2. Roughly speaking,
smoothness of order ss in LpLp is expressed here by the fact that, for
nn large enough,
ωn(f,t)Lpωn(f,t)Lp goes to 0 like
O(ts)O(ts) as t→0t→0.
Clearly Cs=B∞,∞sCs=B∞,∞s when ss is not an integer. It can also
be proved
that when ss is not an integer Ws,p=Bp,psWs,p=Bp,ps. These spaces are different
from one another for integer values of ss, except when p=2p=2 in which case
Hs=B2,2sHs=B2,2s for all values of ss (see [6], p.38).
-
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.
(What does "Endorsed by" mean?)
"This is an introduction to fractional measures of smoothness for functions via Besov spaces that generalize the classical Sobolev smoothness spaces."