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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.

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

  • [1] H. Cartan, Collected works, Springer, Berlin and New York, 1979. MR 540747 (81c:01031)
  • [2] E. Čech, Sur les fonctions continues qui prennent chaque leur valeur un nombre fini de fois, Fund. Math. 17 (1931), 32-39.
  • [3] F. Filipczak, Sur les fonctions continues relativement monotones, Fund. Math. 58 (1966), 75-87. MR 0188381 (32:5820)
  • [4] K. M. Garg, On level sets of a continuous nowhere monotone function, Fund. Math. 52 (1963), 59-68. MR 0143855 (26:1405)
  • [5] M. Laczkovich, Differentiable restrictions of continuous functions, Acta Math. Acad. Sci. Hungar. (to appear). MR 764629 (86d:26010)
  • [6] K. Padmavally, On the roots of equation $ f(x) = \xi $ where $ f(x)$ is real and continuous in $ (a,b)$ but monotonic in no subinterval of $ (a,b)$, Proc. Amer. Math. Soc. 4 (1953), 839-841. MR 0059341 (15:513h)
  • [7] S. M. Ulam, A collection of mathematical problems, Interscience, New York, 1960. MR 0120127 (22:10884)
  • [8] N. H. Williams, Combinatorial set theory, North-Holland, Amsterdam, 1977.
  • [9] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), 63-89. MR 1501735
  • [10] Z. Zahorski, Sur l'ensemble des points singuliers d'une fonction d'une variable réele admittant les dérivés de tous les ordres, Fund. Math. 34 (1947), 183-245. MR 0025545 (10:23c)

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

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