All the above spaces share the common feature that the regularity index is
an integer. In many applications, one is interested to allow
fractional order of smoothness, in order to describe the regularity
of a function in a more precise way. The question thus arises of how to
fill the gaps between integer smoothness classes. There
are at least two instances where such a generalization is very natural:
- In the case of L2L2-Sobolev spaces Hm:=Wm,2Hm:=Wm,2 and when
Ω=RdΩ=Rd,
we can define an equivalent norm based on the Fourier transform, since by
Parseval's formula we have the norm equivalence
∥f∥Hm2∼∫Rd(1+|ω|)2m|f^(ω)|2dω.∥f∥Hm2∼∫Rd(1+|ω|)2m|f^(ω)|2dω.
(1)∥f∥Hs2:=∫Rd(1+|ω|)2s|f^(ω)|2dω,∥f∥Hs2:=∫Rd(1+|ω|)2s|f^(ω)|2dω,
(2) - In the case of CmCm spaces, we note that
supx∈Ω|f(x)-f(x-h)|≤C|h|supx∈Ω|f(x)-f(x-h)|≤C|h| if f∈C1f∈C1 for any
h∈Rdh∈Rd
whereas for an arbitrary function f∈C0f∈C0,
supx∈Ω|f(x)-f(x-h)|supx∈Ω|f(x)-f(x-h)| might go to zero arbitrarily slow as
|h|→0|h|→0.
This motivates the definition of the Hölder spaceCsCs, 0<s<10<s<1 consisting
of those f∈C0f∈C0 such that
supx∈Ω|f(x)-f(x-h)|≤C|h|s.supx∈Ω|f(x)-f(x-h)|≤C|h|s.
(3)supx∈Ω|Δhnf(x)|≤C|h|s,supx∈Ω|Δhnf(x)|≤C|h|s,
(4)∥f∥Cs(Ω):=∥f∥L∞(Ω)+suph∈Rd|h|-s∥Δhnf∥L∞(Ω).∥f∥Cs(Ω):=∥f∥L∞(Ω)+suph∈Rd|h|-s∥Δhnf∥L∞(Ω).
(5)
Let us give two important instances in which the above spaces appear
in a natural way. The first is the study of the restriction of a function
f(x1,⋯,xd)f(x1,⋯,xd) to a manifold of lower dimension, for example
the hyperplane defined by xd=0xd=0. If g(x1,⋯,xd-1)=f(x1,⋯,xd-1,0)g(x1,⋯,xd-1)=f(x1,⋯,xd-1,0)
is such a restriction, then it is known that
f∈Hs(Rdf∈Hs(Rd
for s>12s>12 implies
that
g∈Hs-12(Rd-1)g∈Hs-12(Rd-1).
The second one is the study of the
Brownian motion W(t)W(t) on an interval II, for which it is known that
W(t)W(t) is almost surely in C12-εC12-ε for all ε>0ε>0.
-
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."