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.
- High-order approximation
- Metric linear combinations
- Metric local linear operators
- Set-valued functions
- Set-valued metric divided differences
- Set-valued metric polynomial interpolation