5.2.1.1 The Admissibility Condition

The continuous wavelet transform of a function f (see Chapter 2) is calculated using the formula:

with:

For f in L ² the existence of the coefficients C _f ( a,b) is ensured once ? is in L ². Nonetheless, in order to enable the inversion of the transformation, it is necessary to choose functions ? belonging to L ¹ ? L ² and satisfying the following admissibility condition relating to the Fourier transform of ?:

These hypotheses about the ? function guarantee the possibility of calculating the inverse transform. We then say that ? is an admissible wavelet.

To verify that a real function is a wavelet we use the following simpler sufficient admissibility condition:

The condition ( CS-ad) makes it possible to affirm that the Fourier transform with consequently, the integrals of the condition (CNS-ad) exist.

5.2.1.2 Simple Examples of Admissible Wavelets

The admissibility condition and the condition (CS-ad) are very weak and it is easy to construct wavelets usable for continuous analysis. Almost any function integrating to zero is appropriate, provided that it also verifies some elementary properties.

Let us cite some simple examples of ? functions verifying the...