What Is Variable Bandwidth?

We propose a new notion of variable bandwidth that is based on the spectral subspaces of an elliptic operator Apf where p>0 is a strictly positive function. Denote by c(Ap) the orthogonal projection of A(p) corresponding to the spectrum of A(p) in < subset of>+; 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 =[0,] the main results say that, in a neighborhood of x, a function of variable bandwidth behaves like a band-limited function with local bandwidth (/p(x))1/2. 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 Schrodinger operators. (c) 2017 Wiley Periodicals, Inc.