Subdivision schemes of sets and the approximation of set-valued functions in the symmetric difference metric

TL;DR

This paper develops subdivision schemes for sets in R^n using a new weighted average, enabling approximation of set-valued functions with properties like monotonicity preservation, and extends these methods to general metric spaces.

Contribution

It introduces a novel weighted average of sets for subdivision schemes, allowing for set refinement and approximation in the symmetric difference metric, including interpolation and extrapolation.

Findings

Constructed set-valued subdivision schemes based on a new set average.

Proved monotonicity preservation of the proposed schemes.

Extended the methods to general metric spaces with averaging operations.

Abstract

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.