  In solving scientific, engineering or pure mathematical problems one is often
faced with a need to approximate the function of a given class by the linear
combination of a preferably small number of functions that are localised one
way or another both in the time and frequency domain. Over the last seventy
years or so a range of systems of thus localised functions have been developed
to allow the decomposition and synthesis of functions of various classes. The
most prominent examples of such systems are Gabor functions, wavelets,
ridgelets, curvelets, shearlets and wave atoms. We recently introduced a family
of quasi-Banach spaces -- which we called wave packet spaces -- that
encompasses all those classes of functions whose elements have sparse
expansions in one of the above-mentioned systems, supplied them with Banach
frames and provided their atomic decompositions. Herein we prove that the
Banach frames and sets of atoms of the wave packet spaces are well localised
or, more specifically, that they are near orthogonal. We believe that good
localisation of the wave packet systems -- a property of paramount importance
that can't be taken for granted even for frames in Hilbert spaces, let alone
Banach spaces -- will pave the way, among other things, for their use for
representing Fourier integral operators on Banach spaces by sparse and well
structured matrices by the Galerkin method. This, in its turn, should allow one
to design efficient computer programmes for solving corresponding operator
equations on two-dimensional manifolds.
