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∥=0∥f∥=0 is possible only if f=0f=0; finally, all Cauchy sequences
converge: if we
have a sequence with entries fn∈Cm(I)fn∈Cm(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
Ω∈RdΩ∈Rd,
we define Cm(Ω)Cm(Ω) to be the space of continuous
functions which have bounded and continuous partial derivatives
∂αf:=∂|α|f∂x1α1⋯∂xdαd∂αf:=∂|α|f∂x1α1⋯∂xdαd,
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
f∈Lpf∈Lp with partial derivatives up to order mm in LpLp. Here pp is a
fixed index in [1,+∞][1,+∞]. (Recall that
∥f∥Lp=[∫Ω|f(x)|p]1p∥f∥Lp=[∫Ω|f(x)|p]1p if p<+∞p<+∞ and
∥f∥L∞=supx∈Ω|f(x)|∥f∥L∞=supx∈Ω|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.
-
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."