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Bump Functions with Hölder Derivatives
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- Journal:
- Canadian Journal of Mathematics / Volume 56 / Issue 4 / 01 August 2004
- Published online by Cambridge University Press:
- 20 November 2018, pp. 699-715
- Print publication:
- 01 August 2004
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- Article
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We study the range of the gradients of a
${{C}^{1,\alpha }}$
-smooth bump function defined on a Banach space. We find that this set must satisfy two geometrical conditions: It can not be too flat and it satisfies a strong compactness condition with respect to an appropriate distance. These notions are defined precisely below. With these results we illustrate the differences with the case of 
${{C}^{1}}$-smooth bump functions. Finally, we give a sufficient condition on a subset of 
${{X}^{*}}$ so that it is the set of the gradients of a 
${{C}^{1,1}}$-smooth bump function. In particular, if 
$X$ is an infinite dimensional Banach space with a ${{C}^{1,1}}$-smooth bump function, then any convex open bounded subset of ${{X}^{*}}$ containing 0 is the set of the gradients of a ${{C}^{1,1}}$-smooth bump function.
