Approximation of almost diagonal non-linear maps by lattice Lipschitz operators

TL;DR

This paper introduces a method to approximate almost diagonal nonlinear maps using lattice Lipschitz operators, leveraging interpolation techniques and eigenvector approximations to achieve accurate representations.

Contribution

It develops a novel approximation technique for almost diagonal maps with lattice Lipschitz operators, including explicit error formulas and practical examples.

Findings

Effective approximation of almost diagonal maps achieved

Error bounds and formulas explicitly derived

Illustrative examples demonstrate the method's applicability

Abstract

Lattice Lipschitz operators define a new class of nonlinear Banach-lattice-valued maps that can be written as diagonal functions with respect to a certain basis. In the $n-$dimensional case, such a map can be represented as a vector of size $n$ of real-valued functions of one variable. In this paper we develop a method to approximate almost diagonal maps by means of lattice Lipschitz operators. The proposed technique is based on the approximation properties and error bounds obtained for these operators, together with a pointwise version of the interpolation of McShane and Whitney extension maps that can be applied to almost diagonal functions. In order to get the desired approximation, it is necessary to previously obtain an approximation to the set of eigenvectors of the original function. We focus on the explicit computation of error formulas and on illustrative examples to present our construction.