4.4 Article

High-Order Approximation of Set-Valued Functions

期刊

CONSTRUCTIVE APPROXIMATION
卷 57, 期 2, 页码 521-546

出版社

SPRINGER
DOI: 10.1007/s00365-022-09572-7

关键词

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

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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.
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.

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