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Convergence of best best $ L\sb{\infty }$-approximations


Authors: Abdallah M. Al-Rashed and Richard B. Darst
Journal: Proc. Amer. Math. Soc. 83 (1981), 690-692
MSC: Primary 41A50; Secondary 46E30
DOI: https://doi.org/10.1090/S0002-9939-1981-0630038-9
MathSciNet review: 630038
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Abstract: Let $ (\Omega, \mathcal{A}, \mu)$ be a probability space and let $ \{ {\mathcal{B}_i}\} _{i = 1}^\infty $ be an increasing sequence of subsigma algebras of $ \mathcal{A}$. Let $ A = {L_\infty }(\Omega, \mathcal{A}, \mu)$, let $ {B_i} = {L_\infty }(\Omega ,{\mathcal{B}_i},\mu )$, and let $ f \in A$. Let $ {f_i}$ denote the best best $ {L_\infty }$-approximation to $ f$ by elements of $ {B_i}$. It is shown that $ {\lim _i}{f_i}(x)$ exists a.e.


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

  • [1] T. Andô and I. Amemiya, Almost everywhere convergence of prediction sequence in 𝐿_{𝑝}(1<𝑝<∞), Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 4 (1965), 113–120 (1965). MR 0189077, https://doi.org/10.1007/BF00536745
  • [2] R. B. Darst, Convergence of $ {L_p}$ approximations as $ p \to \infty $, Proc. Amer. Math. Soc. (to appear).
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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0630038-9
Keywords: Best approximation, $ {L_\infty }$
Article copyright: © Copyright 1981 American Mathematical Society