Harmonic Analysis on the Möbius Gyrogroup

Abstract

hool of ehnology nd wngementD olytehni snstitute of veiriD ortugl nd gshwe E genter for eserh nd hevelopment in wthemtis nd epplitionsD heprtment of wthemtisD niversity of eveiroD ortuglF miltonFferreirdipleiriFpt Abstract sn this pper we propose to develop hrmoni nlysis on the oinré ll B n t D model of the nEdimensionl rel hyperoli speF he oinré ll B n t is the open ll of the iuliden nEspe R n with rdius t > 0D entered t the origin of R n nd equipped with wöius dditionD thus forming wöius gyrogroup where wöius ddiE tion in the ll plys the role of vetor ddition in R n . por ny t > 0 nd n ritrry prmeter σ ∈ R we study the (σ, t)EtrnsltionD the (σ, t)EonvolutionD the eigenfunE tions of the (σ, t)EvpleEfeltrmi opertorD the (σ, t)Erelgson pourier trnsformD its inverse trnsform nd the ssoited lnherel9s heoremD whih represent ounE terprts of stndrd toolsD thusD enling n e'etive theory of hyperoli hrmoni nlysisF woreoverD when t → +∞ the resulting hyperoli hrmoni nlysis on B n t tends to the stndrd iuliden hrmoni nlysis on R n , thus unifying hyperoli nd iuliden hrmoni nlysisF es n pplition we onstrut di'usive wvelets on B n t .

Gyrogroup actions: A generalization of group actions

“…Many of classical theorems in group theory continue to hold for gyrogroups, including the Lagrange theorem [13], the fundamental isomorphism theorems [15], and the Cayley theorem [15]. Further, harmonic analysis can be studied in the framework of gyrogroups [4,5], using the gyroassociative law in place of the associative law.…”

Harmonic Analysis on the Einstein Gyrogroup

“…This work is in the same spirit of our previous work [5] where we developed harmonic analysis on the Möbius gyrogroup. The results of this paper can be applied in dierent settings, e.g.…”

On Metric Structures of Normed Gyrogroups

“…In [27], Ungar studies a parametrization of the Lorentz transformation group. This leads to the formation of gyrogroup theory, a rich subject in mathematics [3,9,[23][24][25][26]30]. Loosely speaking, a gyrogroup is a group-like structure in which arXiv:1810.10491v2 [math.MG] 3 Nov 2018 the associative law fails to satisfy.…”