In this work we construct subdivision schemes refining general subsets of R^nand study their applications to the approximation of set-valued functions.Differently from previous works on set-valued approximation, our methods aredeveloped and analyzed in the metric space of Lebesgue measurable sets endowedwith the symmetric difference metric. The construction of the set-valuedsubdivision schemes is based on a new weighted average of two sets, which isdefined for positive weights (corresponding to interpolation) and also when oneweight is negative (corresponding to extrapolation). Using the new average with positive weights, we adapt to sets splinesubdivision schemes computed by the Lane-Riesenfeld algorithm, which requiresonly averages of pairs of numbers. The averages of numbers are then replaced bythe new averages of pairs of sets. Among other features of the resultingset-valued subdivision schemes, we prove their monotonicity preservationproperty. Using the new weighted average of sets with both positive andnegative weights, we adapt to sets the 4-point interpolatory subdivisionscheme. Finally we discuss the extension of the results obtained in the metricspaces of sets, to general metric spaces endowed with an averaging operationsatisfying certain properties.
展开▼
