Set-valued Hermite interpolation

Abstract

The problem of interpolating a set-valued function with convex images is addressed by means of directed sets. A directed set will be visualised as a usually non-convex set in R n consisting of three parts together with its normal directions: the convex, the concave and the mixed-type part. In the Banach space of the directed sets, a mapping resembling the Kergin map is established. The interpolating property and error estimates similar to the point-wise case are then shown; the representation of the interpolant through means of divided differences is given. A comparison to other set-valued approaches is presented. The method developed within the article is extended to the scope of the Hermite interpolation by using the derivative notion in the Banach space of directed sets. Finally, a numerical analysis of the explained technique corroborates the theoretical results.

Metric approximation of set-valued functions of bounded variation

“…First we prove the left inclusion in (12). To show the right inclusion in (12), we use (11) and (9) and write…”

“…Research on approximation and numerical integration of set-valued functions with convex images can be found e.g. in [37,16,36,29,30,32,31,17,9,10,27,7,18,20,8,28,11,6,14]. The standard tools used are the Minkowski linear combinations and the Aumann integral.…”

“…b a k(x)dx = 0, the left inclusion in(13) follows directly from(12). To prove the right inclusion in (13), we start with(11).Denoting R = x∈[a,b] F (x) ∈ K(R d ) we get in view of the second property in (9) )dx co(R), and the right inclusion follows. Note that the middle set in (13) is a weighted average of F (x) on [a, b].…”

Metric Fourier Approximation of Set-Valued Functions of Bounded Variation

“…Research on approximation and numerical integration of set-valued functions with convex images can be found e.g. in [6][7][8][9][10][11]14,[16][17][18]20,[27][28][29][30][31][32]36,37]. The standard tools used are the Minkowski linear combinations and the Aumann integral.…”

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

“…In this approach, sums of numbers in positive operators for real-valued approximation are replaced by Minkowski sums of sets. A more recent approach is to embed the given convex sets into the Banach space of directed sets [3], and to apply any existing method for approximation in Banach spaces [4].…”