5 Wavelet Methods for Pointwise Regularity
Definition 0.1. Suppose that f is defined on IR and let a 1, /3 0
and r 0. f is a chirp of type (a, /?) and regularity r at x$ if there exist
T 0 (actually T = rj~P) and two functions v+ and V- defined and Cr
on [T, +oc[ and indefinitely oscillating such that
(0.4) f(x) =
(x-x0)av+((x-xo)~p)
if x0 x x0+v
(0.5) f(x) \x
xo\av-(\x—XQ|~^)
if XQ T]XXO.
This type of local behavior has been previously considered in physics
and in signal analysis (see [12] and [37]), and this is one of our motiva-
tions for studying chirps. Another motivation is that this local behavior
can be found in some clasical functions of analysis, and we will study in
details such an example in Chapter VII. The third motivation is that the
multifractal formalism for functions may fail because of accumulations
of points where a chirp-like behavior happens (this is implicit in [22]
and this phenomenon will be studied in detail in a forthcoming paper
by A. Arneodo, E. Bacry, S. Jaffard and J-F. Muzy).
In Chapter IV we will show that chirps have a very simple characteri-
zation using the two-microlocal spaces. We will actually characterize in
Theorem 4.2 the sums x(x) + u(x) = f(x) where x(x) is a chirp at XQ
and u)(x) is a C°° function in a neighbourhood of Xo
In Chapter V, we will consider a more precise definition of chirps. We
will study the trigonometric chirps, which have the following asymptotic
expansion.
Definition 0.2. A function f is a trigonometric chirp of type (a,/?)
and regularity r at XQ (a, ft, r 0) if the following expansion holds for
any N1,
(0.6) f(x) = u(x) + \x-xo\av(£\±(x-x0))-(3)+
+ \x-x0\a+f3v(£)(±(x-x0))-(3)
+
+ \x-
x0\^N~W v^-1\±(x
-
x0))-p)
+ \x-Xo\«+WR^\±(x-Xo))-P)
where ± is the sign of x xo, u is C°° in a neighborhood of XQ, and the
v±, 0 q N 17 and ' satisfy the following conditions:
(0.7) the 2N functions are 27r-periodic, belong to the Holder
space Cr+q and f*n v^ (x) dx = 0
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