Localized nonlinear functional equations and two sampling problems in signal processing

TL;DR

This paper develops a theoretical framework for solving nonlinear functional equations in Banach spaces, with applications to nonlinear sampling and signal identification in signal processing.

Contribution

It introduces a strict monotonicity condition ensuring solvability and stability of nonlinear equations, and applies it to sampling and signal identification problems.

Findings

Exponential convergence of Van-Cittert iteration in ll^p spaces.

Framework for solving nonlinear equations with continuous dependence on data.

Application to nonlinear sampling and finite rate of innovation signal identification.

Abstract

Let $1\le p\le \infty$. In this paper, we consider solving a nonlinear functional equation $$f(x)=y,$$ where $x, y$ belong to $\ell^p$ and $f$ has continuous bounded gradient in an inverse-closed subalgebra of ${\mathcal B}(\ell^2)$, the Banach algebra of all bounded linear operators on the Hilbert space $\ell^2$. We introduce strict monotonicity property for functions $f$ on Banach spaces $\ell^p$ so that the above nonlinear functional equation is solvable and the solution $x$ depends continuously on the given data $y$ in $\ell^p$. We show that the Van-Cittert iteration converges in $\ell^p$ with 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.