Ulam-Zahorski problem on free interpolation by smooth functions

Olevskiĭ, A.

doi:10.1090/S0002-9947-1994-1179399-9

Ulam-Zahorski problem on free interpolation by smooth functions
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Trans. Amer. Math. Soc. 342 (1994), 713-727 Request permission

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 ${\left \| g \right \|_{{C^{n + \varepsilon }}}}$)? 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.

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Differentiable restrictions of real functions

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