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Let $f:[1,\infty[$ be a continuous function such that $lim_{x \to > \infty}f(x)=a$. Show that there exist a sequence of function $g_n$ that converge uniformly to $f$ where $g_n(x)=P_n(1/x)$ with $P_n$ a polynomial.

To solve this probleme I think we need to use the approximation of Weierstrass but the fucntion $f$ is not a compact .

I dont know how to proceed and I'm not sure how to use the approximation of Weierstrass theorem.

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Consider the function $g$ on $[0,1]$ given by $g(x) = f(1/x)$ for $x > 0$, $g(0) = a$, and approximate it by polynomials using Weierstrass.

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  • $\begingroup$ can you please explain how do you construct the function g? $\endgroup$ – Elina Dec 18 '16 at 20:48
  • $\begingroup$ What part of it do you not understand? I gave you the formula for $g(x)$, and the value for $x=0$. $\endgroup$ – Robert Israel Dec 18 '16 at 22:27
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Let $g(x)= \begin{cases} f(1/x) & 0<x\leq 1 \\ a & x=0 \end{cases} $

$g(x)$ is continuous on $(0,1]$ because $f(x)$ is a continuous function and also $g(x)$ is continuous at $x=0$. It follow that $g$ is continuous on $[0,1]$.

By the Weierstrass Approximation theorem, there exist a polynomial $P_n(x)$ that converge uniformly to $g(x)$ on $[0,1]$.

Since $\forall x \in[0, \infty)$: $$|g(1/x) - P_n(1/x)| =|f(x) - P_n(1/x)| $$

it follow that $\forall \epsilon >0 , \exists N \in \mathbb{N}$ such that $n>N$:

$$|f(x) - P_n(1/x)|<\epsilon .$$

We conclude that $P_n(1/x)$ converge uniformly to $f(x)$ on $[0,\infty)$.

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