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Localized nonlinear functional equations and two sampling problems in signal processing

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Abstract

Let 1 ≤ p ≤ ∞. In this paper, we consider solving a nonlinear functional equation

f (x) = y,

where x, y belong to ℓpand f has continuous bounded gradient in an inverse-closed subalgebra of ℬ (ℓ2), the Banach algebra of all bounded linear operators on the Hilbert space ℓ 2. We introduce strict monotonicity property for functions f on Banach spaces ℓ pso that the above nonlinear functional equation is solvable and the solution x depends continuously on the given data y in ℓ p. We show that the Van-Cittert iteration converges in ℓ pwith exponential rate and hence it could be used to locate the true solution of the above nonlinear functional equation. We apply the above theory to handle two problems in signal processing: nonlinear sampling termed with instantaneous companding and subsequently average sampling; and local identification of innovation positions and qualification of amplitudes of signals with finite rate of innovation.

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Authors and Affiliations

  1. Department of Mathematics, University of Central Florida, Orlando, FL, 2816, USA

    Qiyu Sun

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  1. Qiyu Sun
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Correspondence to Qiyu Sun.

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Communicated by: Lixin Shen

Research of the author was supported in part by the National Science Foundation (DMS-1109063)

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Sun, Q. Localized nonlinear functional equations and two sampling problems in signal processing. Adv Comput Math 40, 415–458 (2014). https://doi.org/10.1007/s10444-013-9314-3

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