What is Variable Bandwidth?

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 $A_pf = - (pf')'$ where $p>0$ is a strictly positive function. Denote by $c_{\Lambda} (A_p)$ the orthogonal projection of $A_p$ corresponding to the spectrum of $A_p$ in $\Lambda$, the range of this projection is the space of functions of variable bandwidth with spectral set in $\Lambda $. 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 $\Lambda = [0,\Omega] $ the main results say that, in a neighborhood of $x\in R$, a function of variable bandwidth behaves like a bandlimited function with local bandwidth $(\Omega / p(x))^{1/2}$. Although the formulation of the results is deceptively similar to the corresponding results for classical bandlimited functions, the methods of proof are much more involved. On the one hand, we use the oscillation method from sampling theory and frame theoretic methods, on the other hand, we need the precise spectral theory of Sturm-Liouville operators and the scattering theory of one-dimensional Schr\"odinger operators.