What Is Variable Bandwidth?

Abstract

We propose a new notion of variable bandwidth that is based on the spectral subspaces of an elliptic operator where p > 0 is a strictly positive function. Denote by the orthogonal projection of Ap corresponding to the spectrum of Ap in ; the range of this projection is the space of functions of variable bandwidth with spectral set in Λ.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 band‐limited function with local bandwidth .Although the formulation of the results is deceptively similar to the corresponding results for classical band‐limited 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 Schrödinger operators. © 2017 Wiley Periodicals, Inc.