Mediterr. J. Math.
Springer International Publishing 2017
Time–Frequency Concentration, Heisenberg
Type Uncertainty Principles and Localiza-
tion Operators for the Continuous Dunkl
Wavelet Transform on R
Hatem Mejjaoli and Khalifa Trim`eche
Abstract. We consider the continuous Dunkl wavelet transform Φ
sociated with the Dunkl operators on R
. We analyze the concentration
of this transform on sets of ﬁnite measure. In particular, Donoho–Stark
and Benedicks type uncertainty principles are given. Next, we prove
many versions of Heisenberg type uncertainty principles for Φ
titative Shapiro’s dispersion uncertainty principle and umbrella theorem
are proved for the Dunkl continuous wavelet transform. Finally, we in-
vestigate the localization operators for Φ
, in particular we prove that
they are in the Schatten-von Neumann class.
Mathematics Subject Classiﬁcation. Primary 44A05; Secondary 42B10.
Keywords. Dunkl operators, Dunkl wavelet transform, Heisenberg’s type
inequalities, dispersion principle, Shapiro’s theorem, Umbrella theorem,
In the classical setting, the notion of wavelets was ﬁrst introduced by Morlet,
a French petroleum engineer at ELF-Aquitaine, in connection with his study
of seismic traces. The mathematical foundations were given by Grossmann
and Morlet in . The harmonic analyst Meyer and many other mathemati-
cians became aware of this theory and they recognized many classical results
inside it (see [3,23,26]). Classical wavelets have wide applications, ranging
from signal analysis in geophysics and acoustics to quantum theory and pure
mathematics (see [8,18,22] and the references therein).
Next, the theory of wavelets and continuous wavelet transform has been
extended on hypergroups, in particular to the Ch´ebli–Trim`eche hypergroups