What is Variable Bandwidth?
Karlheinz Gr\"ochenig, Andreas Klotz

TL;DR
This paper introduces a new concept of variable bandwidth based on spectral subspaces of an elliptic operator, develops sampling theorems, and establishes density conditions, revealing local bandwidth behavior akin to classical bandlimited functions.
Contribution
It defines a novel variable bandwidth framework using spectral subspaces of elliptic operators and proves sampling and density theorems with advanced spectral analysis techniques.
Findings
Functions of variable bandwidth behave locally like classical bandlimited functions.
Sampling theorems are established for these variable bandwidth functions.
Necessary density conditions are derived in the style of Landau.
Abstract
We propose a new notion of variable bandwidth that is based on the spectral subspaces of an elliptic operator where is a strictly positive function. Denote by the orthogonal projection of corresponding to the spectrum of in , the range of this projection is the space of functions of variable bandwidth with spectral set in . We will develop the basic theory of these function spaces. First, we derive (nonuniform) sampling theorems, second, we prove necessary density conditions in the style of Landau. Roughly, for a spectrum the main results say that, in a neighborhood of , a function of variable bandwidth behaves like a bandlimited function with local bandwidth . Although the formulation of the results is deceptively similar to the corresponding results for…
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