Smooth bump functions and the geometry of banach spaces

By R. Fry, S. McManus. 2002

76 citations recorded by the source corpus.

Abstract We give a brief survey of classical and recent results concerning smooth bump functions on Banach spaces.

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Convex Functions: Constructions, Characterizations and Counterexamples (2010, background): “Building upon results from [178], in [420] it is shown a Banach space is isomorphic to a Hilbert space if X and X ∗ have continuous bump functions that are everywhere twice Gâteaux differentiable (the definition therein stipulated the second derivative must be symmetric).”
Polyfold and Fredholm Theory (2014, background): “The existence of Ck-bump functions on Lp spaces is a consequence of the following result due to Bonic and Frampton, see Theorem 1 in [17].”
Monotone Approximations of Minimum and Maximum Functions and Multi-objective Problems (2012, background): “We also note that other objectives such as arriving in target sets may be formulated using nonnegative smooth bump functions [13].”
Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time (2017, unknown): “Fry & McManus (2002)) function ζ (t − t0) = exp T m+n−1 ( 1−µ1(t−t0)m+n−1 ) , (10) with the properties that ζ (0) = 1 and ζ (T ) = 0.”