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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Convexity conditions and intersections with smooth functions

Authors: S. Agronsky, A. M. Bruckner, M. Laczkovich and D. Preiss
Journal: Trans. Amer. Math. Soc. 289 (1985), 659-677
MSC: Primary 26A51; Secondary 26A48
MathSciNet review: 784008
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Abstract: A continuous function that agrees with each member of a family $ \mathcal{F}$ of smooth functions in a small set must itself possess certain desirable properties. We study situations that arise when $ \mathcal{F}$ consists of the family of polynomials of degree at most $ n$, as well as certain larger families and when the small sets of agreement are finite. The conclusions of our theorems involve convexity conditions. For example, if a continuous function $ f$ agrees with each polynomial of degree at most $ n$ in only a finite set, then $ f$ is $ (n + 1)$-convex or $ (n + 1)$-concave on some interval. We consider also certain variants of this theorem, provide examples to show that certain improvements are not possible and present some applications of our results.

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Keywords: Monotonicity, $ n$-convexity
Article copyright: © Copyright 1985 American Mathematical Society