Wavelet as affine group, practical evidence and theoretical hypothesis. Olga Lucía Quintero Montoya; Doctor en Ingeniería de Sistemas de Control, Universidad Nacional de San Juan, Argentina. Marzo 28 de 2016.
Speaker: Olga Lucía Quintero Montoya
Studies: Doctor en Ingeniería de Sistemas de Control, Universidad Nacional de San Juan, Argentina
Affiliation: GRIMMAT – Research group in mathematical modeling, EAFIT University
Talk title: Wavelet as affine group, practical evidence and theoretical hypothesis
The coherent states associated with the ax+b-group, which are now called wavelets, were first constructed by Alasken and Kauder (1968, 1969), as the result of the action of operators U(a,b) on a given function f. This results on the fact that any function f in L2(R) can be approximated by wavelets superposition. It happens, that these results can summarized on the fact that the continuous wavelet transform can map isometrically the L2(R) in a subspace H subset of Hilbert space. And also, if we choose a subset of L2(R), H2 (Hardy) we can interpreter the same transform as a isometry from H2 to Bergman space and any function of this Bergman space is associated to a function in H2. In this case, result obvious (for some people), that its values of certain discrete families of points completely determine the function.
The group structure proposed by Alasken and Klauder was not exploited so much, because of people went to the discretely labeled wavelet families and these do not correspond to subgroups of the ax+b-group. However we will discuss why it is fascinating to me to find a relationship between the continuous representation theory using the affine group (that inherits the singularity spectrum obtained by continuous wavelet transform and Holder exponent calculus) and the capability of the discrete approximations via multiresolution analysis to describe a dynamical behavior of a system.