The Real and Complex Techniques in Harmonic Analysis from the Point of View of Covariant Transform

This note reviews complex and real techniques in harmonic analysis. We describe a common source of both approaches rooted in the covariant transform generated by the affine group. Keywords: wavelet, coherent state, covariant transform, reconstruction formula, the affine group, ax+b-group, square integrable representations, admissible vectors, Hardy space, fiducial operator, approximation of the identity, maximal functions, atom, nucleus, atomic decomposition, Cauchy integral, Poisson integral, Hardy--Littlewood maximal functions, grand maximal function, vertical maximal functions, non-tangential maximal functions, intertwining operator, Cauchy-Riemann operator, Laplace operator, singular integral operator, SIO, boundary behaviour, Carleson measure.