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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On approximate antigradients


Authors: Xiao-Xiong Gan and Karl R. Stromberg
Journal: Proc. Amer. Math. Soc. 119 (1993), 1201-1209
MSC: Primary 26B35; Secondary 26B05, 41A30, 41A63
MathSciNet review: 1169878
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Abstract: For $ n \in \mathbb{N}$ and $ I = [0,1]$, let $ {I^n}$ be the unit cube and $ {\lambda ^n}$ the Lebesgue measure in $ {\mathbb{R}^n}$. It is proved that if $ f:{I^n} \to {\mathbb{R}^n}$ and $ {F_0}:{I^n} \to \mathbb{R}$ are continuous and $ \varepsilon > 0$, then there exist a continuous $ F:{I^n} \to \mathbb{R}$ and an open set $ W \subset {({I^n})^ \circ }$ with $ {\lambda ^n}(W) = 1$ such that

(i) $ \nabla F$ exists and is continuous on $ W$,

(ii) $ \vert\vert\nabla F(x) - f(x)\vert\vert < \varepsilon \;\forall x \in W$, and

(iii) $ \vert F(x) - {F_0}(x)\vert < \varepsilon \;\forall x \in {I^n}$, where $ \vert\vert y\vert\vert = {\left( {\sum\nolimits_{j = 1}^n {y_j^2} } \right)^{1/2}}\;\forall y \in {\mathbb{R}^n}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1169878-7
PII: S 0002-9939(1993)1169878-7
Keywords: Antigradient
Article copyright: © Copyright 1993 American Mathematical Society