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# 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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### A density proof not via Stone-Weierstrass

Context: I have proved Weierstrass' theorem (polynomials are dense in $C[a,b]$) in two ways: one using Bernstein polynomials, and one using convolutions. You can also use Stone-Weierstrass theorem, ...
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### How does Weierstrass' theorem follow from Mergelyan's theorem?

According to Theorems 1 and 3 in this review article we have Weierstrass: Suppose $f$ is a continuous function on a closed bounded interval $[a,b] \subset\mathbb{R}$. For every $\epsilon > 0$ ...
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### Complex analysis - $\bar{z}$ cannot be uniformly approximated by polynomials in $z$ on the closed unit disc in $\mathbb{C}$.

In reference to this question, I tried to prove that $\bar{z}$ cannot be uniformly approximated by complex polynomials in $z$ on the closed unit disc $D$. I came up with a proof, but I'm not entirely ...
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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 ...
392 views

### 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 ...
36 views

### 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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### Uniform approximation by even polynomial

Proposition Let $\mathcal{P_e}$ be the set of functions $p_e(x) = a_o + a_2x^2 + \cdots + a_{2n}x^{2n}$, $p_e : \mathbb{R} \to \mathbb{R}$ Show that all $f:[0,1]\to\mathbb{R}$ can be ...
488 views

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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### 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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### There is a sequence $(P_{n})$ of polynomials such that $P_{n}(\cos x) \to f(x)$ uniformly over $[0,\pi]$.

Show that for any function continuous $f:[0,\pi] \to \mathbb{R}$, there is a sequence $(P_{n})$ of polynomials such that $$P_{n}(\cos x) \to f(x)\;\text{uniformly over}\;[0,\pi].$$ The Weierstrass ...
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### Are of Polynomials of Continuous functions complete?

The Stone-Weierstrass Theorem guarantees that every continuous function defined on a closed interval [a,b] can be approximated as closely as desired by a polynomial. I'm curious in polynomials of ...
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### $\int_0^1 t^{n\lambda}f(t)dt=0$ for all but finitely many $n$

Let $\lambda\in\mathbb{R}$ be fixed and positive. Let $f$ be continuous on $[0,1]$ satisfy $\int_0^1 t^{n\lambda}f(t)dt=0$ for all but finitely many $n\in\mathbb{N}$. Prove $f=0$, for all $x\in[0,1]$. ...
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### Assume $f$ is continuous on $[-1,1]$. If $\int_{-1}^{1}f(x)x^ndx=0$, for all nonnegative integer $n$, can we prove that $f(x)=0$?

Assume $f$ is continuous. If $$\int_{-1}^{1}f(x)x^ndx=0$$ holds for every non-negative integer $n$ , can we prove $f(x)=0$ for all $|x|\le 1$ ? Is it possible to have $f(x)x^n$ odd?
### Prove that there exists a function $g: [0,1] \to \mathbb{R}$ which is 1-Lipschitz, satisfies $g (0) = 0$ and has the following property.
Question. Consider a continuous function $f: [0,1] \to \mathbb {R}$. Prove that there exists a function $g: [0,1] \to \mathbb{R}$ which is 1-Lipschitz, satisfies $g (0) = 0$ and has the property ...
Let $a, b \in \mathbb{R}, a<b$, and let $f:[a,b]\to \mathbb{R}$ be a continuous function. Given $\varepsilon>0$, show that there exists $\alpha_1, \alpha_2, \dots, \alpha_n \in \mathbb{R}$ and \$...