Box-constrained monotone L_∞-approximations and Lipschitz-continuous regularized functions
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Let f:[0,1]→[0,1] be a nondecreasing function. The main goal of this work is to provide a regularized version, say f̃_L, of f. Our choice will be a best L_∞-approximation to f in the set of functions h:[0,1]→[0,1] which are Lipschitz-continuous, for a fixed Lipschitz norm bound L, and verify the boundary restrictions h(0)=0 and h(1)=1. Our findings allow to characterize a solution through a monotone best L_∞-approximation to the Lipschitz regularization of f. This is seen to be equivalent to follow the alternative way of the average of the Pasch-Hausdorff envelopes. We include results showing stability of the procedure as well as directional differentiability of the L_∞-distance to the regularized version. This problem is motivated within a statistical problem involving trimmed versions of distribution functions as to measure the level of contamination discrepancy from a fixed model.
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