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Continuity of best approximants


Authors: D. Landers and L. Rogge
Journal: Proc. Amer. Math. Soc. 83 (1981), 683-689
MSC: Primary 46E30; Secondary 41A65
DOI: https://doi.org/10.1090/S0002-9939-1981-0630037-7
MathSciNet review: 630037
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {C_n}$, $ n \in {\mathbf{N}}$, be $ \Phi $-closed lattices in an Orlicz-space $ {L_\Phi }(\Omega, \mathcal{A}, \mu)$ and assume that $ {C_n}$ increases or decreases to a $ \Phi $-closed lattice $ {C_\infty }$. Let $ {f_n}$, $ n \in {\mathbf{N}}$, be $ \mathcal{A}$-measurable real valued functions with $ {f_n} \to f\mu $-a.e. and $ \sup \vert{f_n}\vert \in {L_\Phi }$. If $ {g_n}$ is a best $ \Phi $-approximant of $ {f_n}$ in $ {C_n}$ it is shown that $ {\underline {\lim } _{n \in {\mathbf{N}}}}{g_n}$ and $ {\overline {\lim } _{n \in {\mathbf{N}}}}{g_n}$ are best $ \Phi $-approximants of $ f$ in $ {C_\infty }$.


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

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0630037-7
Keywords: Best approximants, $ \sigma $-lattices, conditional expectations, characterization
Article copyright: © Copyright 1981 American Mathematical Society

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