Convergence of best best $L_{\infty }$-approximations
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- by Abdallah M. Al-Rashed and Richard B. Darst PDF
- Proc. Amer. Math. Soc. 83 (1981), 690-692 Request permission
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
- T. Andô and I. Amemiya, Almost everywhere convergence of prediction sequence in $L_{p}\,(1<p<\infty )$, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 4 (1965), 113–120 (1965). MR 189077, DOI 10.1007/BF00536745 R. B. Darst, Convergence of ${L_p}$ approximations as $p \to \infty$, Proc. Amer. Math. Soc. (to appear).
- Walter Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. MR 0344043
Additional Information
- © Copyright 1981 American Mathematical Society
- 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