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What Is Variable Bandwidth?
Author(s) -
Gröchenig Karlheinz,
Klotz Andreas
Publication year - 2017
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21694
Subject(s) - mathematics , linear subspace , bandwidth (computing) , sampling theory , variable (mathematics) , mathematical analysis , pure mathematics , computer science , telecommunications , sample size determination , statistics
We propose a new notion of variable bandwidth that is based on the spectral subspaces of an elliptic operatorA p f = − d d x( p ( x ) d d x) f where p  > 0 is a strictly positive function. Denote byc 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 band‐limited function with local bandwidth( Ω / 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 Schrödinger operators. © 2017 Wiley Periodicals, Inc.

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