What Is Variable Bandwidth?

Abstract

Abstract. We propose a new notion of variable bandwidth that is based on the spectral subspaces of an elliptic operatorf where p > 0 is a strictly positive function. Denote by c Λ (A p ) the orthogonal projection of A p corresponding to the spectrum of A p in Λ ⊂ R + , 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 Λ = [0, Ω] the main results say that, in a neighborhood of x ∈ R, a function of variable bandwidth behaves like a bandlimited function with local bandwidth (Ω/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 Schrödinger operators.

Density of sampling and interpolation in reproducing kernel Hilbert spaces

“…We will need the following elementary facts about frames and Riesz sequences in a reproducing kernel Hilbert space H. They are copied from [24,37]. Lemma 3.9.…”

“…Proof. The proof from [24,37] is included for completeness. Let P V be the orthogonal projection on the subspace V of H. Inequality The proof of (24) is the same, except that P V = I for frames and thus equality holds in the last step.…”

“…(ii) In [24] we treated a unitarily equivalent model of the Paley-Wiener space and studied a new concept of variable bandwidth. In that case the density results are formulated differently and also involve a different geometry.…”

“…(v) Sampling in spaces of bandlimited functions on Lie groups [20,39]. (vi) Sampling in spaces of variable bandwidth [24]. (vii) Sampling in spaces of bandlimited functions associated to an integral transform, for example, the Hankel transform [2].…”

Function spaces obeying a time-varying bandlimit

“…This makes it difficult to make the concept of a timevarying bandlimit precise. For several approaches in the literature, see, for example, [6,7,8,9,10,11].…”