Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Time-dependent wavelets, alias space-time affine coherent states

Jean-Pierre Antoine

Published Jan 1, 1998• 1 citation
OpenAlex

Abstract

We describe several families of wavelets associated to the Galilei group, extended by space and time dilations. The construction follows a general method based on group representations which are square integrable only on a homogeneous space of the underlying group.

Related papers

Galilean wavelets: Coherent states of the affine Galilei group

Jean-Pierre Antoine, I. Mahara

We derive Galilean wavelets, by which we mean coherent states of the affine Galilei group, that is, the Galilei group extended by independent space and time dilations. The construction follows a general method based on square integrable group representations, possibly modulo a subgroup, i.e., on a homogeneous space of the underlying group. We also examine the restriction to the Schrödinger subgroup, which contains only dilations that leave invariant the Schrödinger and the heat equations.

Journal of Mathematical Physics•Nov 1, 1999•8 citations

Higher-dimensional wavelets

Jean-Pierre Antoine, Romain Murenzi, Pierre Vandergheynst +1 more

In the previous chapters, we have thoroughly discussed the 2-D CWT and some of its applications. Then we have made the connection with the group theoretical origins of the method, thus establishing a general framework, based on the coherent state formalism. In the present chapter, we will apply the same technique to a number of different situations involving higher dimensions: wavelets in 3-D space ℝ 3 , wavelets in ℝ n ( n > 3), and wavelets on the 2-sphere S 2 . Then, in the next chapter, we will treat time-dependent wavelets, that is, wavelets on space–time, designed for motion analysis. In all cases, the technique is the same. First one identifies the manifold on which the signals are defined and the appropriate group of transformations acting on the latter. Next one chooses a square integrable representation of that group, possibly modulo some subgroup. Then one constructs wavelets as admissible vectors and derives the corresponding wavelet transform. Three-dimensional wavelets Some physical phenomena are intrinsically multiscale and three-dimensional. Typical examples may be found in fluid dynamics, for instance the appearance of coherent structures in turbulent flows, or the disentangling of a wave train in (mostly underwater) acoustics, as discussed above. In such cases, a 3-D wavelet analysis is clearly more adequate and likely to yield a deeper understanding [56].

Cambridge University Press eBooks•Sep 23, 2004•0 citations

Coherent states associated to group representations which are square integrable on a homogeneous space

Jean-Pierre Antoine

•Jan 1, 1991•0 citations

COHERENT STATES AND THEIR GENERALIZATIONS: A MATHEMATICAL OVERVIEW

S. Twareque Ali, Jean-Pierre Antoine, Jean‐Pierre Gazeau +1 more

We present a survey of the theory of coherent states (CS) and some of their generalizations, with emphasis on the mathematical structure, rather than on physical applications. Starting from the standard theory of CS over Lie groups, we develop a general formalism, in which CS are associated to group representations which are square integrable over a homogeneous space. A further step allows us to dispense with the group context altogether, and thus obtain the so-called reproducing triples and continuous frames introduced in some earlier work. We discuss in detail a number of concrete examples, namely semisimple Lie groups, the relativity groups and various types of wavelets. Finally we turn to some physical applications, centering on quantum measurement and the quantization/dequantization problem, that is, the transition from the classical to the quantum level and vice versa.

Reviews in Mathematical Physics•Oct 1, 1995•149 citations

Coherent States, Wavelets And Applications G24 - Lln

Jean‐Pierre Gazeau, Richard Kerner, Jean-Pierre Antoine +2 more

Although coherent states and wavelets have developed by now into full-fledged subdisciplines, with plenty of applications in physics, mathematics and engineering, both have deep roots in group theory,. The construction of coherent states was identified as a group-theoretical problem in the pioneering work of Perelomov and Gilmore, whereas wavelets are simply the coherent states associated to the affine group of the line (or the similitude group of the plane for the two-dimensional case). It was therefore appropriate to have both topics represented in the GROUP-24 Colloquium.

CRC Press eBooks•Nov 30, 2003•0 citations

TheSU(1,1) coherent states related to the affine group wavelets

J. Bertrand, Mich�le Irac-Astaud

Czechoslovak Journal of Physics•Dec 1, 2001•2 citations
In the previous chapters, we have thoroughly discussed the 2-D CWT and some of its applications. Then we have made the connection with the group theoretical origins of the method, thus establishing a general framework, based on the coherent state formalism. In the present chapter, we will apply the same technique to a number of different situations involving higher dimensions: wavelets in 3-D space ℝ 3 , wavelets in ℝ n ( n > 3), and wavelets on the 2-sphere S 2 . Then, in the next chapter, we will treat time-dependent wavelets, that is, wavelets on space–time, designed for motion analysis. In all cases, the technique is the same. First one identifies the manifold on which the signals are defined and the appropriate group of transformations acting on the latter. Next one chooses a square integrable representation of that group, possibly modulo some subgroup. Then one constructs wavelets as admissible vectors and derives the corresponding wavelet transform. Three-dimensional wavelets Some physical phenomena are intrinsically multiscale and three-dimensional. Typical examples may be found in fluid dynamics, for instance the appearance of coherent structures in turbulent flows, or the disentangling of a wave train in (mostly underwater) acoustics, as discussed above. In such cases, a 3-D wavelet analysis is clearly more adequate and likely to yield a deeper understanding [56].
We present a survey of the theory of coherent states (CS) and some of their generalizations, with emphasis on the mathematical structure, rather than on physical applications. Starting from the standard theory of CS over Lie groups, we develop a general formalism, in which CS are associated to group representations which are square integrable over a homogeneous space. A further step allows us to dispense with the group context altogether, and thus obtain the so-called reproducing triples and continuous frames introduced in some earlier work. We discuss in detail a number of concrete examples, namely semisimple Lie groups, the relativity groups and various types of wavelets. Finally we turn to some physical applications, centering on quantum measurement and the quantization/dequantization problem, that is, the transition from the classical to the quantum level and vice versa.
Although coherent states and wavelets have developed by now into full-fledged subdisciplines, with plenty of applications in physics, mathematics and engineering, both have deep roots in group theory,. The construction of coherent states was identified as a group-theoretical problem in the pioneering work of Perelomov and Gilmore, whereas wavelets are simply the coherent states associated to the affine group of the line (or the similitude group of the plane for the two-dimensional case). It was therefore appropriate to have both topics represented in the GROUP-24 Colloquium.