On strong unicity of $L_{1}$-approximation
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- by András Kroó PDF
- Proc. Amer. Math. Soc. 83 (1981), 725-729 Request permission
Abstract:
Let ${C_1}$ be the space of continuous functions on $[0,1]$ with norm $\left \| f \right \| = \int _0^1 {|f(x)|} dx$, 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 $\left \| {f - \tilde g} \right \| \leqslant \left \| f \right \| + \delta$ it follows that $\left \| {\tilde g} \right \| \leqslant Kw_f^* (\delta )$, 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.)References
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Additional Information
- © Copyright 1981 American Mathematical Society
- 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