Classically harmonic analysis is concerned with the properties of Fourier transforms and Fourier series, in particular convergence of summability methods, the treatment of convolution operators and the applications to complex analysis, partial differential equations and approximation theory.
Modern harmonic analysis has also enlarged its views to encompass analytical problems posed on a Lie group or a homogeneous space instead of Rn. Nilpotent Lie groups and symmetric spaces provide the most natural context in this regard.
On the more applied side, one of the most successful outputs of Fourier analysis of the last ten years is the theory of wavelets, which has become an important branch of numerical and applied mathematics. Besides these applied aspects, wavelets present many interesting theoretical aspects related to approximation theory, to harmonic analysis in phase space and to square integrable representations of Lie groups.