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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Proof of inequality in Weierstrass approximation theorem proof through probability

I found this exercise in my probability theory book. The problem guides you through a proof of the Weierstrass Approximation Theorem through probability theory. My question is only about part b, so ...
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Convergence rate of multivariate polynomials.

Given a continuous function $f:[a,b]\rightarrow\mathbb{R}$, it is a well known result that, for each $n\geq0$, there exists a polynomial $p_n$ that best approximates $f$ in $\|\cdot\|_\infty$ among ...
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Weierstrass approximation theorem - absolute value

I am really struggling on this exercise, and don't even know where to start. (a) Use the fact that $|a|=\sqrt{a^2}$ to prove that, given $\epsilon>0$, there exists a polynomial $q(x)$ satisfying ...
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Describe the intervals where $f_n(x)$ converges uniformly

Problem: $f_n(x) = \frac{6}{(1+x^{2n})}$ and $x\in\mathbb{R}$. Find all real numbers $x$ where $f_n(x)$ converges and describe the limit function. I found the limit function to be f(x)=\...
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Approximation by smooth functions by changing values at arbitrarily small interval

Assume $f\in C[0,1]$ is smooth (i.e. infinitely many times differentiable) on $(0,\frac 12)$ and $(\frac 12,1)$. Let $\epsilon>0$ be arbitrarily small. Can we approximate $f$ in supremum norm by ...
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Is the set of all polynomials dense in the set of all continuous functions that maps from a compact set to reals

Let $C$ be a set of all continuous functions that map from a compact set to the reals. Is the set of all polynomials dense in $C$, with respect to the sup metric? Can I show that by using the Stone-...
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A problem about the Weierstrass approximation theorem

Let $\alpha\gt 0$ .Using the Weierstrass Theorem, prove that every continuous function $f(x)$ on [0,$\infty$] with $\lim_{x \to \infty} f(x) = 0$ can be uniformly approximated as closely as we like by ...
Let $f:[0,1]\rightarrow\mathbb{R}$ be continuous such that $\int_0^1 x^{2k}f(x)dx=0$
Let $f:[0,1]\rightarrow\mathbb{R}$ be continuous such that $\int_0^1 x^{2k}f(x)dx=0$, $\forall k\geq 1$, integers. Show that $f(x)=0$, $\forall x\in[0,1]$. *Let $y=x^2$, then this function can be ...
Polynomials in $y=e^x$ dense in C[1,e]
I try to understand a proof of Theorem that I found in Tran van Thuong article. \ Theorem: For each integer $N\ge1$, the set of functions: {$e^{nx} : n \ge N$} has a linear span dense in $C[0,1]$. ...