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Approximation of set-valued functions is introduced and discussed under a convexity assumption. In particular, a theorem of Korovkin type is derived for a class of approximation methods. (Author)

Generalizations of some known results on the best, best linear and best one-sided approxima- tions by trigonometric polynomials of the classes of 2\pi - periodic functions presented in the form of convolutions to the case of set-valued functions are obtained

We study the approximation of univariate and multivariate set-valued functions (SVFs) by the adaptation to SVFs of positive samples-based approximation operators for real-valued functions. To this end, we introduce a new weighted average of several sets and study its properties. The approximation results are obtained in the space of Lebesgue measurable sets with the symmetric difference metric. In particular, we apply the new average of sets to adapt to SVFs the classical Bernstein approximation operators, and obtain a set-valued analog of the Weierstrass approximation theorem. The rate of approximation of H\"older continuous SVFs by Bernstein operators is studied and shown to be asymptotically equal to that for real-valued functions. Finally, the results obtained in the metric space of sets are generalized to metric spaces endowed with an average satisfying certain properties.

A new continuity for set-valued functions is introduced, and an existence theorem is proved for such continuous set-valued functions.

In this work we construct subdivision schemes refining general subsets of R^n and study their applications to the approximation of set-valued functions. Differently from previous works on set-valued approximation, our methods are developed and analyzed in the metric space of Lebesgue measurable sets endowed with the symmetric difference metric. The construction of the set-valued subdivision schemes is based on a new weighted average of two sets, which is defined for positive weights (corresponding to interpolation) and also when one weight is negative (corresponding to extrapolation). Using the new average with positive weights, we adapt to sets spline subdivision schemes computed by the Lane-Riesenfeld algorithm, which requires only averages of pairs of numbers. The averages of numbers are then replaced by the new averages of pairs of sets. Among other features of the resulting set-valued subdivision schemes, we prove their monotonicity preservation property. Using the new weighted average of sets with both positive and negative weights, we adapt to sets the 4-point interpolatory subdivision scheme. Finally we discuss the extension of the results obtained in the metric spaces of sets, to general metric spaces endowed with an averaging operation satisfying certain properties.