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

Abstract

In solving scientific, engineering or pure mathematical problems one is frequently faced with a need to approximate the function of a given class with a specified precision by the linear combination of a preferably small number of simpler functions. This can often achieved by choosing the simpler functions localised one way or another both in the time and frequency domain. Constructing a set of linearly independent functions, let alone a basis, with a given time-frequency localisation is a formidable and often unsolvable problem, though. A much better chance one stands in building a set of time-frequency localised functions that constitutes a so-called frame – a generalisation of the notion of the basis, whose elements need not be linear independent, rather than a basis.

Over the last seventy years or so, a range of frames have been designed 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 frames, supplied them with Banach frames – the kind of frames that ensure that any element of the class of functions for which a frame was designed can be decomposed and reconstructed using that frame – and provided their atomic decomposition. Herein we prove that the Banach frames for and sets of atoms of the wave packet spaces – which we call $wave$ $packet$ $systems$ – are indeed localised in the time and frequency domain or, more specifically, that they are near orthogonal; and therefore so are all of the above-mentioned examples of frames.

We shall also show that, unlike those examples, the wave packet system can be made to assume a wide range of types and degrees of time-frequency localisation by the suitable choice of values of the parameters of the system. This, we believe, makes the wave packet systems not only suitable for decomposing, synthesising or approximating functions of a wide range of quasi-Banach function spaces in an efficient and effective way, but also for their use for representing linear bounded operators on the quasi-Banach spaces by sparse and well structured matrices using the Galerkin method. This, in its turn, should allow one to design efficient computer programs for solving corresponding operator equations on two-dimensional manifolds using finite element methods.