Stephane Jaffard and Yves Meyer
belong to the Holder space
and are indefinitely
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
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) =
(/(x0 + Aft) - P(Xh)) +
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
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 ).
The differentiability of a(x) at the rationals which are ratios of two
odd numbers was established by J. Gerver in 1970 in . 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) . In Chapter