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

Abstract

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.