  We describe recent nonlinear analytic approximation tools in the classical
setting of Hardy spaces in the upper half plane and show how to transfer them
to the higher dimensional real setting of harmonic functions in upper half
spaces. It is known [6] that all harmonic functions in higher dimensions are
combinations of holomorphic functions on 2 dimensional planes, extended as,
constant in normal directions. We derive representation theorems, with
corresponding isometries, opening the door for applications in higher
dimensions, to the processing of highly oscillatory multidimensional signals.
