High-Order Approximation of Set-Valued Functions

We introduce the notion of metric divided differences of set-valued functions. With this notion we obtain bounds on the error in set-valued metric polynomial interpolation. These error bounds lead to high-order approximations of set-valued functions by metric piecewise-polynomial interpolants of high degree. Moreover, we derive high-order approximation of set-valued functions by local metric approximation operators reproducing high-degree polynomials.

Automated summary

We introduce the concept of metric divided differences for set-valued functions, which allows us to obtain bounds on the error in set-valued metric polynomial interpolation. These error bounds lead to high-order approximations of set-valued functions using high-degree metric piecewise-polynomial interpolants. Additionally, we derive high-order approximations of set-valued functions using local metric approximation operators that reproduce high-degree polynomials.