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Set-valued interpolation. (English) Zbl 1128.26018

Bayreuth: Univ. Bayreuth, Fakultät für Mathematik und Physik (Dissertation 2006). Bayreuther Math. Schr. 79, vi, 154 p. (2007).
This 154 pages booklet contains the PhD dissertation presented by the author at the University of Bayreuth in Germany. The subject of the dissertation is a “directed sets approach” of the polynomial interpolation of compact convex valued multi-functions, accompanied by corresponding numerical procedures and their software implementation on some particular cases.
The first three chapters are introductory and present the necessary notations, definitions and preliminary results from the so-called “set-valued analysis”; a special attention is accorded to the definition and basic properties of the positively linear, order preserving and isometric embedding of the cone, \({\mathcal C}(\mathbb R^n)\) of compact convex subsets of \(\mathbb R^n\) into the ordered Banach space, \(\vec {\mathcal D}^n\), of directed sets. Chapter 3 contains definitions and results concerning the directed derivative of a compact convex-valued multi-function, introduced via the usual derivative of the corresponding embedding into the Banach space of directed sets, and which turns out to be fully characterized by the usual derivatives of the support functions of certain associated multi-functions.
The core of the dissertation, laying down its theoretical framework, is presented in Chapters 4 and 5, containing the definitions and basic properties of the divided differences for set-valued mappings, including the Hermite-Genocchi formula and, respectively, the set-valued polynomial interpolation, the remainder formulae and error estimates in the framework of directed derivatives. Chapter 6 contains some numerical applications, including numerical tests and graphics for a number of particular examples while the last chapter contains a short description of the software implementation (SVUPI).

MSC:

26E25 Set-valued functions
49J53 Set-valued and variational analysis
41A10 Approximation by polynomials
58C06 Set-valued and function-space-valued mappings on manifolds
47H04 Set-valued operators
26-02 Research exposition (monographs, survey articles) pertaining to real functions