# Questions tagged [weierstrass-approximation]

If $f$ is a continuous real-valued function on $[a,b]$ and if any $\epsilon>0$ is given, then there exists a polynomial $p$ on $[a,b]$ such that $$|f(x)-p(x)|<\epsilon$$ for all $x$ in $[a,b]$. In words, any continuous function on a closed and bounded interval can be uniformly approximated on that interval by polynomials to any degree of accuracy.

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### Prove that there exists a sequence of polynomials $(p_n)$ that converges uniformly to $f$ on $[a,b]$

I've been stuck on the following problem for a some time but have it started somewhat and not sure whether it is correct or how to proceed. Let $f$ be a continuous function on $[a,b]$. Prove that ...
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### $h$ is strictly monotone iff polynomials in $h$ are uniformly dense

Suppose $h$ is continuous on $[0,1]$. Then, I have to prove the following: $h$ is strictly monotone iff every continuous function on $[0,1]$ can be uniformly approximated on $[0,1]$ by a polynomial ...
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I have some questions regarding the sequences of functions in the Weierstrass M-test: Weierstrass M-test:. Suppose that $\{f_n\}$ is a sequence of real- or complex-valued functions defined on a set $... 2answers 60 views ### Show that there exist a sequence of function$g_n$that converge uniformly to$f$Let$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)$with$...
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Let f be a continuous function on [0, 1].The moments of f can refer to the quantities $\int_0^1x^nf(x)dx$ for n = 0,1,2... . Prove that two continuous functions defined on [0, 1] are identical if ...
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### $\int_I x^n f(x)dx=0$ and $I$ open imply $f=0$

Let $I$ be a bounded interval of $\mathbb{R}$ and $f$ be a continuous function on $I$. Assume $I$ is closed, ie: $I=[a,b]$, and $\forall n\geq 0$, $\int_I x^n f(x)dx=0$. Then, one can show with ...
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### How to solve the non-linear ordinary differential equation $\frac{d^2 y}{dx^2}=y^2$?

Mathematica gives the following result: $6^{1/3} \times$ WeierstrassP[$\frac{x + C_1}{6^{1/3}}$, {$0, C_2$}], where $C_1 \& C_2$ are constants of integration. It is possible to find the series ...
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### Proof of Peano's existence theorem for ODEs

Theorem: Given a continuous function $f:(a,b)\times (c,d)\to \mathbb R$, the problem $$\begin{cases} y'(t)=f(t,y(t))\\ y(t_0)=y_0\end{cases}$$ has a local solution in some neighbourhood of a given ...
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