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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On strong unicity of $ L\sb{1}$-approximation


Author: András Kroó
Journal: Proc. Amer. Math. Soc. 83 (1981), 725-729
MSC: Primary 41A52
DOI: https://doi.org/10.1090/S0002-9939-1981-0630044-4
MathSciNet review: 630044
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Abstract: Let $ {C_1}$ be the space of continuous functions on $ [0,1]$ with norm $ \left\Vert f \right\Vert = \int_0^1 {\vert f(x)\vert} dx$, and let $ G \subset {C_1}$ be a finite dimensional unicity subspace, i.e. any $ f \in {C_1}$ possesses a unique best approximation out of $ G$. Consider an arbitrary $ f \in {C_1}$ such that 0 is its best approximation. Then for any $ 0 < \delta < {\delta _0}$ and $ \tilde g \in G$ with $ \left\Vert {f - \tilde g} \right\Vert \leqslant \left\Vert f \right\Vert + \delta $ it follows that $ \left\Vert {\tilde g} \right\Vert \leqslant Kw_f^* (\delta )$, where $ w_f^*(h) = {(w_f^{ - 1}(h)h)^{ - 1}}$ and $ {w_f}(h)$ denotes the modulus of continuity of $ f$. ($ - 1$ is used to denote the inverse function.)


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DOI: https://doi.org/10.1090/S0002-9939-1981-0630044-4
Article copyright: © Copyright 1981 American Mathematical Society