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If a sequence of polynomials converge uniformly in $\mathbb{R}$ to $f$, is $f$ necessarily a polynomial?

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  • $\begingroup$ Your question is subtle as the domain of convergence. A taylor series of $\sin(x)$, $\cos(x)$ and $e^x$, for exemple, converge uniformly in compact intervals of $\mathbb{R}$ and pontise in all $\mathbb{R}$. That is, fixed in any compact interval $[a,b]$ in $\mathbb{R}$ the the series converges uniformly. $\endgroup$ – MathOverview Jan 11 '13 at 11:39
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Claim: If $\{P_n\}$ is a sequence of polynomials which converges to $0$ uniformly on an unbounded subset $S$ of the real line, then $P_n$ is constant for $n$ large enough.

Indeed, $\sup_{x\in S}|P_n(x)|<1$ for $n$ large enough, hence $P_n$ is bounded on $S$. Fix such a $n$. Let $x_k\in S$, $|x_k|\to +\infty$. If $P_n$ is of degree $d$, then $1>|P_n(x_k)|\geqslant |c_nx_k^d|$, where $2c_n$ is the leading term. It enforces $d=0$.

Now, we apply the preceding result to $P_{n+1}-P_n$.

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Yes. Any two functions $f,g$ with $\sup|f(x)-g(x)|<\infty$ must grow at the same rate. A degree $d$ polynomial $g$ grows at rate $O(|x|^d)$. So if polynomials $g_1,g_2,\dots$ converge uniformly to $f$ then all of $f,g_1,g_2,\dots$ grow at rate $O(|x|^d)$ for some integer $d$, and thus $g_1,g_2,\dots$ must have degree $d$. By Lagrange interpolation, for each $j$ the $j$'th coefficient of $g_k$ converges as $k\to\infty$ to some $c_j$, and $f$ is the polynomial with these limits $c_j$ as coefficients.

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  • $\begingroup$ MacQuillan, there is any reference book for it result? $\endgroup$ – MathOverview Jan 11 '13 at 11:43

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