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Proceedings of the American Mathematical Society

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Convergence of $ L\sb{p}$ approximations as $ p\rightarrow \infty $


Author: Richard B. Darst
Journal: Proc. Amer. Math. Soc. 81 (1981), 433-436
MSC: Primary 41A50; Secondary 41A65, 46E30
MathSciNet review: 597657
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Abstract: Let $ (\Omega, \mathcal{A}, \mu)$ be a probability space and let $ \mathcal{B}$ be a subsigma-algebra of $ \mathcal{A}$. Let $ A = {L_\infty }(\Omega, \mathcal{A}], \mu)$ and let $ B = {L_\infty }(\Omega ,\mathcal{B},\mu )$. Let $ f \in A$, and for $ 1 < p < \infty $, let $ {f_p}$ denote the best $ {L_p}$ approximation to $ f$ by elements of $ {L_p}(\Omega ,\mathcal{B},\mu )$. It is shown that $ {\lim _{p \to \infty }}{f_p}$ exists a.e. The function $ {f_\infty }$ defined by $ {f_\infty }(x) = {\lim _{p \to \infty }}{f_p}(x)$ is a best $ {L_\infty }$ approximation to $ f$ by elements of $ B:\vert\vert f - f_\infty\vert\vert _\infty = \inf \{ \vert\vert f - g\vert\vert _\infty ; g \in B \}$. Indeed, $ {f_\infty }$ is a best best $ {L_\infty }$ approximation to $ f$ by elements of $ B$ in the sense that for each $ E \in \mathcal{B}$ the restriction, $ {f_\infty }\vert E$, of $ {f_\infty }$ to $ E$ is a best $ {L_\infty }$ approximation to the restriction, $ f\vert E$, of $ f$ to $ E$. Since there is at most one best best $ {L_\infty }$ approximation to $ f$, $ {f_\infty }$, is the best best $ {L_\infty }$ approximation to $ f$ by elements of $ B$.


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DOI: https://doi.org/10.1090/S0002-9939-1981-0597657-X
Keywords: Best approximation, $ {L_\infty }$, $ {L_p}$-space, probability space
Article copyright: © Copyright 1981 American Mathematical Society