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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34 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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79 views

Specific proof technique of the complex Stone-Weierstrass theorem

The question is as follows: If $f:\mathbb{T}\rightarrow\mathbb{C}$ is continuous, prove that there is a sequence of polynomials $p_n(z,\bar{z})$ such that $p_n\rightarrow f$ uniformly for every $z\...
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166 views

Prove Weierstrass Theorem using Fejer Theorem

Using the following theorem: The trigonometric polynomials ($\mathbb{C} \to \mathbb{C}$) are uniformly dense in $C(\mathbb{T})$ (functions $\mathbb{C} \to \mathbb{C}$ that are continuous and periodic ...
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79 views

Limit Point 2D Bolzano-Weierstrass

I'm struggling with the following problem: a) Let A be the set of complex numbers. A number z is called, as in the real case, a limit point, the limit point of the set A if for every $\epsilon>0$, ...
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1answer
556 views

Help proving the Weierstrass Approximation Theorem using Fejer's Theorem

I found a series of steps designed to give a constructive proof of WAT using Fejer's Theorem. For clarity, I'm using the following statement of WAT: Let $[a,b]\subset \mathbb{R}$ suppose that $f:[a,...
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31 views

Smooth approximation (under supremum norm) of distance to algebraic set in $\mathbb{R}^n$.

Given a set $S$ which is the zeroes of a finite number of homogenous polynomials in $x\in\mathbb{R}^n$, I want a constant $\alpha$ and a $C^2$ approximation, denoted $d$, to the function $d(x,S)=\inf_{...
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59 views

Stone-Weierstrass applications

I’m currently trying to familiarize myself with the Stone-Weierstrass theorem and its applications. When browsing Wikipedia, I found the following: If X and Y are two compact Hausdorff spaces and f : ...
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274 views

Use Stone-Weierstrass theorem to show that $f$ will be $0$ for all $x\in [0,1]$.

Fix a real number $\lambda>0$. Let $f\in C[0,1]$ satisfy $\int_0^1 t^{n\lambda}f(t)dt=0$ for all but finitely many $n\in\mathbb{N}$. What can be deduced about the function $f$? Claim: $f$ will be $...
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462 views

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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1answer
62 views

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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1answer
60 views

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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315 views

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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1answer
149 views

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 ...
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112 views

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 ...
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147 views

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 $C[0,1]$. ...