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

TL;DR
This paper demonstrates that the Banach frames and atoms of wave packet spaces are nearly orthogonal, which enhances their potential for efficient representation and computation of Fourier integral operators in Banach spaces.
Contribution
The paper proves near orthogonality of Banach frames and atoms in wave packet spaces, advancing their application in sparse representations and operator computations.
Findings
Banach frames and atoms are well localized and nearly orthogonal.
Wave packet spaces encompass various function classes with sparse expansions.
Potential for efficient numerical methods for operator equations on manifolds.
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…
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Taxonomy
TopicsMathematical Analysis and Transform Methods · Advanced Harmonic Analysis Research · Advanced Numerical Analysis Techniques
