6
Stephane Jaffard and Yves Meyer
(0.8)
R^}
belong to the Holder space
Cr+N
and are indefinitely
oscillating.
We will show that the trigonometric chirps are also characterized, up
to a C°° function in the neighborhood of xo, by simple conditions on
their continuous wavelet transform.
A less oscillatory behavior appears when / has some selfsimilarity.
Suppose for instance that XQ = 0 and that 0 a 1; / is selfsimilar of
order a at the origin if there exists a A 1 such that
/(Ax) =
Xaf(x).
This definition is often too rigid in applications, hence the following
modification (which also works for arbitrarily large values of a).
A function / is approximately selfsimilar at XQ if it satisfies (for some
a 0, A 1 and polynomial P)
f(x0 + h)- P(h) =
\~a
(/(x0 + Aft) - P(Xh)) +
o(\h\a).
This equality is clearly implied by the following equation (0.9).
Definition 0.3. Let a, 7 0, and 0 A 1; a function f is a
logarithmic chirp of order (a, A) and of regularity 7 at xo if there exist
two functions G+ and G- in C7(IR) such that for some polynomial P,
(0.9) f(x0 + h)- P(h) = |fc|aG±(log(±/i)) + o(\hn
where ± is the sign of h, and G+ and G- are (log A)-periodic.
In Chapter VI we will give sharp necessary conditions and sufficient
conditions on the continuous wavelet transform of / for the existence of
logarithmic chirps.
We will study the Riemann function in detail in Chapter VII, because
it is a very nice example of a function that exhibits chirps and logarithmic
chirps (partial results concerning these properties were first investigated
by J.J. Duistermaat in [10]).
The differentiability of a(x) at the rationals which are ratios of two
odd numbers was established by J. Gerver in 1970 in [13]. Several new
proofs of this result were obtained later, one of them by M. Holschneider
and P. Tchamitchian using a wavelet analysis of a(x) [18]. In Chapter
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