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In solving scientific, engineering or pure mathematical problems one is oftenfaced with a need to approximate the function of a given class by the linearcombination of a preferably small number of functions that are localised oneway or another both in the time and frequency domain. Over the last seventyyears or so a range of systems of thus localised functions have been developedto allow the decomposition and synthesis of functions of various classes. Themost prominent examples of such systems are Gabor functions, wavelets,ridgelets, curvelets, shearlets and wave atoms. We recently introduced a familyof quasi-Banach spaces -- which we called wave packet spaces -- thatencompasses all those classes of functions whose elements have sparseexpansions in one of the above-mentioned systems, supplied them with Banachframes and provided their atomic decompositions. Herein we prove that theBanach frames and sets of atoms of the wave packet spaces are well localisedor, more specifically, that they are near orthogonal. We believe that goodlocalisation of the wave packet systems -- a property of paramount importancethat can't be taken for granted even for frames in Hilbert spaces, let aloneBanach spaces -- will pave the way, among other things, for their use forrepresenting Fourier integral operators on Banach spaces by sparse and wellstructured matrices by the Galerkin method. This, in its turn, should allow oneto design efficient computer programmes for solving corresponding operatorequations on two-dimensional manifolds.

On near orthogonality of the Banach frames of the wave packet spaces