Next
talk of the Seminar (15:30, Room
11.1.12)
21/09/2022 “Quaternion Hyperbolic Fourier Transforms”
Milton Ferreira (School of Technology and Management Polytechnic of Leiria &
CIDMA)
During the last decades,
quaternion Fourier transforms (QFT) have been deeply investigated and found
applications to color image processing, nuclear magnetic resonance imaging,
speech recognition, among others. The two most wellknown forms are the
twosided QFT and the rightsided QFT. Several properties, uncertainty
principles, timefrequency distributions were studied for these QFT by
several authors.
In this seminar, we present the
hyperbolic counterpart of the QFT. We show their main properties, inversion
formula, Plancherel and Parseval’s Theorems.
Concerning the uncertainty principles, we show a sharp Pitt’s inequality
for the twosided QFT that allows deriving a logarithmic uncertainty
principle, and Weyl’s–Heisenberg uncertainty principle in our context. Donoho–Stark’s uncertainty principle and Benedick’s
qualitative uncertainty principle are also given together with a hyperbolic
Poisson summation formula. These results depend heavily on the properties
of gyrogroups that we will introduce and explain.
In the limiting case, we recover
all the results of the QFT in the Euclidean case.
This Seminar is supported in part by the Portuguese Foundation for
Science and Technology (FCT  Fundação para a Ciência e a Tecnologia),
through CIDMA  Center for Research and Development in Mathematics and
Applications, within project UIDB/04106/2020.
