4
Stephane Jaffard and Yves Meyer
s G IN; see [29]). We will pay special attention to the "limit cases" where
s = ^. More precisely, we will consider the two following problems :
Prove similar results for functions in BV(TR)
Find which Sobolev spaces contain the characteristic function of a
given set Q (depending on the regularity of the boundary 9Q).
In Section II.4, we show how wavelet methods can extend the usual
trace formulas for functions in Sobolev or Besov spaces; we illustrate
the ideas involved with an example. Let / be a function defined on
IR2
and which belongs to the Sobolev space
HS(1R2).
Consider now the one
variable functions fa which are the traces of / on the line x = a; then
fa G iy
s _ 1 / / 2
(IR) (if s 1/2) uniformly in the parameter a; and for
almost every a, fa G
H8(JR).
Our purpose is to obtain intermediate
results. For instance, here, if s 1/2
sf
s, fa G
Hs
(IR) except
on a set of values of the parameter a of Hausdorff dimension at most
l-2(s-s').
In Chapter III we examine the relation between wavelet expansions
and lacunary trigonometric series, returning first to the problem of the
"almost characterization" of differentiability of a function. In
Chapter I different criteria of differentiability are given; we study here
a specific lacunary trigonometric series whose regularity cannot be ob-
tained through microlocal arguments and we show how direct estimates
give some information on the pointwise behavior of this trigonometric
series.
Up to now we described the singularity of f{x) at x
0
only by the
Holder exponent a(x
0
) so that, if a 1, we looked for the order of
magnitude of \f(x) /(xo)| when x tends to xo, without taking into
account the oscillations of f(x) f(xo). One of our purposes is to inves-
tigate some specific local behaviors such as approximate selfsimilarities
and very strong oscillatory behaviors ("chirps" of the form
xa
sin(l/a;^))
and show that the two-microlocal spaces allow us to characterize and eas-
ily distinguish these behaviors. The definition of a chirp requires first
the notion of indefinitely oscillating functions.
A function g(x) (x G IR) is indefinitely oscillating if the iterated prim-
itives of g (defined by #o = 9 and p
n +
i(x ) = f 9n(t) dt) are 0(1) at
infinity. We can now define the chirps of type (a, (5) and of regularity r.
a is the Holder exponent as above, whereas /3 measures the oscillations
which accelerate when x tends to XQ. Finally, r 0 gives information
on the global smoothness of / .
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