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