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Monotone $ L\sb 1$-approximation on the unit $ n$-cube

Authors: Richard B. Darst and Robert Huotari
Journal: Proc. Amer. Math. Soc. 95 (1985), 425-428
MSC: Primary 41A52; Secondary 41A29
MathSciNet review: 806081
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Abstract: Let $ \Omega $ be the unit $ n$-cube $ {[0,1]^n}$, and let $ M$ be the set of all real-valued functions on $ \Omega $, each of which is nondecreasing in each variable separately. If $ f:\Omega \to \mathbb{R}$ is continuous, we show that there exists an (essentially) unique, best $ {L_1}$-approximation, $ {f_1}$, to $ f$ by elements of $ M$, and that $ {f_1}$ is continuous.

References [Enhancements On Off] (What's this?)

  • [1] R. B. Darst and R. Huotari, Best $ {L_1}$-approximation of bounded, approximately continuous functions on $ [0,1]$ by nondecreasing functions, J. Approx. Theory 43 (1985), 178-189. MR 775785 (87g:41063)
  • [2] D. Landers and L. Rogge, Natural choice of $ L_{1}$-approximants, J. Approx. Theory 33 (1981), 268-280. MR 647853 (83f:41021)

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