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}$.

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

  • [1] Stanislaw Saks, Theory of the integral, Stechert, New York, 1937.
  • [2] Karl R. Stromberg, Introduction to classical real analysis, Wadsworth International, Belmont, Calif., 1981. Wadsworth International Mathematics Series. MR 604364
  • [3] Paul R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950. MR 0033869
  • [4] Edwin Hewitt and Karl Stromberg, Real and abstract analysis, Springer-Verlag, New York-Heidelberg, 1975. A modern treatment of the theory of functions of a real variable; Third printing; Graduate Texts in Mathematics, No. 25. MR 0367121

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Keywords: Antigradient
Article copyright: © Copyright 1993 American Mathematical Society