DEPARTMENT OF MECHANICS AND APPLIED MATHEMATICS
WAVELET COEFFICIENTS OF FUNCTIONS IN SIGNAL AND IMAGE PROCESSING
HEAD OF THE PROJECT Jüri LIPPUS, Ph.D.
DESCRIPTION
We study the coefficients of wavelet and multiresolution-type expansions of functions with a given majorant of the modulus of continuity.
The generalized Lipschitz classes of continuous functions are defined in the following way. We say that the function
(
)()(0) = 0(1 + 2)
(1) + (2);C)(f,)C = O(())() = 
(0 < 
1)
APPLICATION
The problem has its applications in signal and image processing, particularly in detecting edges while enhancing images. It is well known that in regions of slow change of the function its wavelet coefficients decay rapidly while in the regions where it changes rapidly, the speed of decrease of the coefficients is slow.
Y. Meyer found necessary and sufficient coefficient criteria for wavelet and multiresolution-type expansions of usual Lipschitz classes. S. Jaffard studied the relations between the local behaviour of the function and its wavelet coefficients. Analogous problems were also studied by M. Holschneider, who generalized these results to the case where the majorant is of the form
() = log
MAIN RESULT
Our main result in this direction is that the coefficient criteria of ordinary Lipschitz classes hold for a slightly larger class of majorants, namely those, satisfying the so-called Bari-Stechkin condition. This generalizes the results of Y. Meyer and M. Holschneider.
Analogous problems in a slightly different context have been studied by the author earlier. We have also proved some localized variants of the above result that take into account the behaviour of the function in a small neighbourhood of a given point.
20/04/1998 webmaster