Localized nonlinear functional equations and two sampling problems in signal processing

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.