Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



Ulam-Zahorski problem on free interpolation by smooth functions

Author: A. Olevskiĭ
Journal: Trans. Amer. Math. Soc. 342 (1994), 713-727
MSC: Primary 26A48; Secondary 26A51, 41A05
MathSciNet review: 1179399
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let f be a function belonging to $ {C^n}[0,1]$. Is it possible to find a smoother function $ g \in {C^{n + 1}}$ (or at least $ {C^{n + \varepsilon }}$) which has infinitely many points of contact of maximal order n with f (or at least arbitrarily many such points with fixed norm $ {\left\Vert g \right\Vert _{{C^{n + \varepsilon }}}}$)? It turns out that for n = 0 and 1 the answer is positive, but if $ n \geq 2$, it is negative. This gives a complete solution to the Ulam-Zahorski question on free interpolation on perfect sets.

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

  • [ABLP] S. Agronsky, A. M. Bruckner, M. Laczkovich, and D. Preiss, Convexity conditions and intersections with smooth functions, Trans. Amer. Math. Soc. 298 (1985), 659-677. MR 784008 (86g:26012)
  • [Ag] S. Agronsky, Intersections of continuous functions with families of smooth functions, Real Anal. Exchange 10 (1984/85), 25-30. MR 795599
  • [Br] J. Brown, Differentiable restrictions of real functions, Real Anal. Exchange 13 (1987/88), 44-45.
  • [Bu] Z. Buczolich, Sets of convexity of continuous function, Acta Math. Hungar. 52 (1988), 291-303. MR 991864 (90c:26036)
  • [Ca] H. Cartan, Collected works, Springer-Verlag, Berlin and New York, 1979. MR 540747 (81c:01031)
  • [Fe] H. Federer, Geometric measure theory, Springer, New York, 1969. MR 0257325 (41:1976)
  • [GPY] J. Goodman, J. Pach, and C. Yap, Mountain climbing, ladder moving, and the ring-width of a polygon, Amer. Math. Monthly 96 (1989), 494-510. MR 999412 (90h:52010)
  • [La] M. Laczkovich, Differentiable restrictions of continuous functions, Acta Math. Hungar. 44 (1984), 355-360. MR 764629 (86d:26010)
  • [Ol] A. Olevskiĭ, Some interpolation problems in real and harmonic analysis, Real Anal. Exchange 16 (1990/91), 363-372. MR 1087503 (92d:41007)
  • [Sc] The Scottish Book, Mathematics from the Scottish café, Birkhäuser, Basel and Boston, 1981. MR 666400 (84m:00015)
  • [St] E. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, NJ, 1970. MR 0290095 (44:7280)
  • [U] S. Ulam, A collection of mathematical problems, Interscience, New York, 1960. MR 0120127 (22:10884)
  • [Z] Z. Zahorski, Sur l'ensemble des points singulière d'une fonction d'une variable réele admettand des dérivées de tous les ordres, Fund. Math. 34 (1947), 183-245. MR 0025545 (10:23c)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 26A48, 26A51, 41A05

Retrieve articles in all journals with MSC: 26A48, 26A51, 41A05

Additional Information

Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society