Approximations of Set-Valued Functions by Metric Linear Operators

Abstract

In this work, we introduce new approximation operators for univariate setvalued functions with general compact images. We adapt linear approximation methods for real-valued functions by replacing linear combinations of numbers with new metric linear combinations of finite sequences of compact sets, thus obtaining "metric analogues" operators for set-valued functions. The new metric linear combination extends the binary metric average of Artstein. Approximation estimates for the metric analogue operators are derived. As examples we study metric Bernstein operators, metric Shoenberg operators and metric polynomial interpolants.

Multi-segmental representations and approximation of set-valued functions with 1D images

“…The property is the existence of a continuous selection through any point of the graph of a CBV multifunction. It is an extension of a result by Hermes [8] on the existence of a continuous selection of a CBV multifunction, and the proof is based on our construction of metric chains in [5]. A similar construction is used by Chistyakov [9] to prove a similar result but with CBV selections.…”

“…Proof. For a fixed ( x, y) ∈ Graph(F), x ∈ [a, b], y ∈ R n we construct "chains" as in [5]. Let x i = a + i h, i = 0, .…”

Computation of the Metric Average of 2D Sets with Piecewise Linear Boundaries

“…In this work we present an algorithm for the computation of the metric average of 2D sets with piecewise linear boundaries. Our algorithm provides a computational method for the approximation of set valued functions with images in R 2 from a finite number of samples, by techniques based on the metric average [3][4][5].…”

“…In [3], the metric average is used for piecewise linear approximation of set-valued functions. Extending these results, Dyn et al applied the metric average to the approximation of set-valued functions from a finite number of their samples, using spline subdivision schemes and more generally positive approximation operators [4,5]. These methods are based on repeated computations of the metric average.…”

Subdivision Schemes of Sets and the Approximation of Set-Valued Functions in the Symmetric Difference Metric

“…The convergence and the approximation results obtained in [10] are based on the metric property of the metric average. The adaptation to sets of certain positive linear operators based on the metric average is described in [16]. For reviews on set-valued approximation methods see also [13,27].…”