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Let $x_n$ be a increasing sequence of negative real numbers that converge to $0$. Let $f$ be a function defined on $x_n$ such that $f(x_n)$ is a increasing sequence of negative numbers that converge to zero. Can I find a $C^\infty$ or real-analytic function $g$ defined on a ball around zero such that $g(x_n)=f(x_n)$ for all $n$ larger than some positive integer?

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No. Take $x_n=-\frac 1{n^2}$ and $f(x_n)=-\frac 1n$. –  fedja Jul 22 '11 at 3:06
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3 Answers 3

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As fedja says in the comments, the answer is "No" in general. However, there is a known condition on when it is possible to interpolate a sequence by a smooth function. This can be found in Section 2.8 of Chapter I of Kriegl and Michor's The Convenient Setting for Global Analysis (though it is not said where the concept originates from).

The set up is as follows: we have a locally convex topological vector space $E$, and a sequence $(x_n)$ in $E$. We say that $(x_n)$ converges fast or falls fast to $x \in E$ if, for each $k \in \mathbb{N}$, the sequence $n^k(x_n - x)$ is bounded. Then the result is:

Special Curve Lemma: Let $(x_n)$ be a sequence which converges fast to $x$ in $E$. Then the infinite polygon through the $(x_n)$ can be parameterised as a smooth curve $c \colon \mathbb{R} \to E$ such that $c(1/n) = x_n$ and $c(0) = x$.

Although this is only an "if", it shouldn't be hard to check whether the "only if" holds or not. For a quick proof of the above, see Kriegl and Michor's book (p16 in the printed version; it's free online via the above link).

This uses the notion of smoothness in LCTVS's that Kriegl and Michor describe (in great detail) in their book. For finite dimensional vector spaces, it is the same as the usual one.

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If you forgive an off topic humorous aside, I notice from the book's website that the FREE pdf download apparently masses 4.68 milligrams. I wonder how they worked that out? –  Harald Hanche-Olsen Jul 22 '11 at 10:25
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Let me remark on the $C^m$ version of the question for $m \geq 1$. A classic result of H. Whitney from

Differentiable functions defined in closed sets I, Transactions A.M.S. 36 (1934), 369–387

reads as follows: Suppose that $f : \lbrace x_1, x_2 ,\cdots\rbrace \rightarrow \mathbb{R}$ is given, with $x_k$ some convergent increasing sequence. Suppose that the $k$'th divided difference quotient based at $x_n$, defined inductively by

$$ \Delta_{n,n+k+1} f : = \frac{\Delta_{n,n+k} f - \Delta_{n+1,n+k+1} f}{x_{n} - x_{n+k+1}} $$ and $$\Delta_{n,n} f = f(x_n)$$ satisfies $|\Delta_{n,n+k+1} f| \leq \Lambda$ uniformly in $n \geq 1$ and $0 \leq k \leq m$ for some $\Lambda < \infty$. In addition, suppose that $$\Delta_{n,n+k+1} f \rightarrow c_k$$ as $n \rightarrow \infty$ for each $0 \leq k \leq m$. Then there exists a $C^m$ function $g: \mathbb{R} \rightarrow \mathbb{R}$ with $g(x_n) = f(x_n)$ for all $n \geq 1$ and

$$\|g\|_{C^m} \leq C(m) \Lambda.$$

In Whitney's original paper, this theorem was proven in greater generality for $f : E \rightarrow \mathbb{R}$ and $E \subset \mathbb{R}$ arbitrary and closed instead of the special case where $E$ consists of a sequence with a single limit point.

I believe that a similar constructive characterization is known for the $C^\infty$ case under the assumption that $\lbrace x_k\rbrace$ is quickly decreasing in the sense that $P(k) |x_k| \rightarrow 0$ for every polynomial $P(k)$. Even for the example of $x_n=-1/n$ I do not know the answer for a general $f$. For quickly decreasing sequences $\lbrace x_k \rbrace$ a recent result used in work of Charles Fefferman and Fulvio Ricci gives a characterization for which functions $f: \lbrace x_k \rbrace \rightarrow \mathbb{R}$ can be extended into $C^\infty(\mathbb{R})$. Unfortunately, I learned of this at a recent conference and I cannot locate the preprint. I am not sure of the relationship between their result and Andrew's answer, since polygonal curves are never graphs of $C^\infty$ functions (or even $C^1$ functions).

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The following paper answers this question for functions of a real variable under weaker assumptions that the Special Curve Lemma mentioned above: MR1245559 (94k:26024) Frölicher, Alfred ; Kriegl, Andreas . Differentiable extensions of functions. Differential Geom. Appl. 3 (1993), no. 1, 71--90.

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