show/hide this revision's text 3 Fixed some wording and corrected statement of Whitney's Theorem

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 also not sure of the relationship between their result and Andrew's answer, since polygonal curves do not arise as are never graphs of $C^\infty$ functions (or even $C^1$ functions).
show/hide this revision's text 2 added 121 characters in body

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 also not sure of the relationship between their result and Andrew's answer, since polygonal curves do not arise as graphs of $C^\infty$ functions (or even $C^1$ functions).
show/hide this revision's text 1

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 $\Lambda < \infty$. 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 also not sure of the relationship between their result and Andrew's answer, since polygonal curves do not arise as graphs of $C^\infty$ functions (or even $C^1$ functions).