Best finite constrained approximations of one-dimensional probabilities

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Chuang Xu, Arno Berger

This paper studies best finitely supported approximations of one-dimensional probability measures with respect to the L r -Kantorovich (or transport) distance, where either the locations or the weights of the approximations’ atoms are prescribed. Necessary and sufficient optimality conditions are established, and the rate of convergence (as the number of atoms goes to infinity) is discussed. In view of emerging mathematical and statistical applications, special attention is given to the case of best uniform approximations (i.e., all atoms having equal weight). The approach developed in this paper is elementary; it is based on best approximations of (monotone) L r -functions by step functions, and thus different from, yet naturally complementary to, the classical Voronoi partition approach.

Original languageEnglish
JournalJournal of Approximation Theory
Pages (from-to)1-36
Number of pages36
Publication statusPublished - 2019

    Research areas

  • Asymptotically best approximation, Balanced function, Best uniform approximation, Constrained approximation, Kantorovich distance, Quantile function


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