Uncertainty principles in Banach spaces and signal recovery

Abstract

A very general uncertainty principle is given for operators on Banach spaces. Many consequences are derived, including uncertainty principles for Bessel sequences in Hilbert spaces and for integral operators between measure spaces. In particular it implies an uncertainty principle for Lp (G), 1 ≤ p ≤ ∞, for a locally compact Abelian group G, concerning simultaneous approximation of f ∈ Lp (G) by gf and H & f for suitable g and H. Taking g and over(H, ^) to be characteristic functions then gives an uncertainty principle about ε{lunate}-concentration of f and over(f, ^), which generalizes a result of Smith, which in turn generalizes a well-known result of Donoho and Stark. The paper also generalizes to the setting of Banach spaces a related result of Donoho and Stark on stable recovery of a signal which has been truncated and corrupted by noise. In particular, this can be applied to the recovery of missing coefficients in a series expansion. © 2006 Elsevier Inc. All rights reserved.