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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300 views

Explanation of Stone-Weierstrass theorem's proof

Theorem: Let $A \subset C(K)$ such that $A$ is a subalgebra with unity $1$ For each $x, y \in K $ with $x \neq y $, there exists $f \in A$ such that $f(x)\neq f(y)$. Then $ \...
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Proof explanation of Stone-Weierstrass theorem

> Theorem: Let $A \subset C(K)$ such that $A$ is a subalgebra with unity $1$ For each $x, y \in K $ with $x \neq y $, there exists $f \in A$ such that $f(x)\neq f(y)$. Then $ \...
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250 views

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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3answers
120 views

$h$ 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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1answer
542 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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0answers
450 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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2answers
334 views

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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Could you please help give a hint to solve this problem in real analysis (uniform convergence)? [closed]

$(a)$ Show that every continuous function $f$ on $[a,b]$ is the uniform limit of polynomials of the form $p_n(x^3)$. $(b)$ Describe the subspace of $C[-1,1]$ functions which are uniform limits of ...
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2answers
333 views

Polynomial approximation (Weierstrass' Theorem) with equality at the endpoints

Let $f:\mathbb R \rightarrow\mathbb R $ be continuous, and choose $-\infty < a<b<\infty$ and $\epsilon > 0$. Show there exists a polynomial $p$ such that: a) $p(a)=f(a),\; p(b)=f(b)$ b) $...
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1answer
263 views

The Series of functions in Weierstrass M test

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 $...
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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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2answers
470 views

Problem about Weierstrass Approximation

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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3answers
90 views

$\int_I x^n f(x)dx=0$ 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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1answer
84 views

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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3answers
2k views

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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1answer
155 views

Showing a 2D continuous function can be approximated by a finite sum of “simpler” continuous functions

Show that given a real-valued continuous function $f$ on $[0,1] \times [0,1]$ and an $\epsilon > 0,$ there exist real-valued continuous functions $g_1, \dots, g_n$ and $h_1 , \dots, h_n$ on $[0,...
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1answer
687 views

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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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
54 views

Using Weierstrass to prove certain limit: Carothers Ch.11 q.10

Let $(x_i)$ be a sequence of numbers in $(0,1)$ such that the $lim_{n→∞} (1/n \sum_{i=1}^{n} x_i^k)$ exists for every $k=0,1,2,....$ Show that $lim_{n→∞} (1/n \sum_{i=1}^{\infty} f(x_i))$ exists for ...
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611 views

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 ...