Title

Localized Nonlinear Functional Equations And Two Sampling Problems In Signal Processing

Keywords

Average sampling; Banach algebra; Infinite matrix; Instantaneous companding; Nonlinear functional equation; Shift-invariant space; Signal with finite rate of innovation; Strict monotonicity·Inverse-closedness

Abstract

Let 1 ≤ p ≤∞. In this paper, we consider solving a nonlinear functional equation f (x) = y, where x, y belong to ℓp and 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 ℓ2.We introduce strict monotonicity property for functions f on Banach spaces ℓpso that the above nonlinear functional equation is solvable and the solution x depends continuously on the given data y in ℓp. We show that the Van-Cittert iteration converges in ℓpwith 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.

Publication Date

4-1-2014

Publication Title

Advances in Computational Mathematics

Volume

40

Issue

2

Number of Pages

415-458

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1007/s10444-013-9314-3

Socpus ID

84881117810 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84881117810

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