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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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7
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2answers
347 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, ...
4
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1answer
87 views

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 $\epsilon > 0$ ...
4
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1answer
768 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 ...
4
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3answers
126 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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0answers
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 ...
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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 ...
3
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1answer
194 views

Show the existence of a sequence of polynomials that converges to $f$ on any compact subset of $\mathbb{R}$.

If $f$ is continuous function ($f:\mathbb{R}\rightarrow\mathbb{R}$), then there exists a sequence of polynomials which converges to $f$ on any compact subset of $\mathbb{R}$. Proof: (Weierstrass) If $...
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1answer
48 views

What is the set of pointwise limits of polynomials?

The set of pointwise limits of continuous functions from from $\mathbb{R}$ to $\mathbb{R}$ is the set of Baire class 1 functions. My question is, my question is, what is the set of pointwise limits ...
2
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2answers
350 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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2answers
688 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 ...
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2answers
488 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 ...
2
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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 ...
2
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1answer
403 views

Proving Peano's Existence Theorem by approximating with $C^{\infty}$ functions using Weierstrass' Theorem.

Let $f:B_r(x_0)\to\mathbb{R}^n$ be continuous. Prove there always exists a local solution $x:[0,\delta)\to\mathbb{R}^n$ satisfying $$x(0)=x_0, \hspace{1cm} x'(t)=f(x(t)), \quad \forall t \in (0,\...
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1answer
284 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 $...
2
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1answer
334 views

Weierstrass approximation theorem. Approximation of |x|.

I'm studying a proof of the Weierstrass approximation theorem that requires an uniform approximation using polynomials of the function |x| in the interval $[-1,1]$ i.e. we need a sequence of ...
2
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1answer
65 views

Approximating $f(x,y)$ as $g(x/y)$

Specific problem So here is my problem. I have a function $$ f(x,y\,|\,\alpha) = \frac{x(\alpha y^2 - 1)}{y(\alpha x^2 - 1)} $$ were x,y and $\alpha$ correspond to some physical parameters and thus I ...
2
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1answer
40 views

$X:=\{z \in \mathbb C : |z|\le 1\}$ ; is $\{p(z,\bar z) | p(x,y) \in \mathbb R[x,y] \}$ dense in $C(X , \mathbb C) $ under sup metric?

Let $X:=\{z \in \mathbb C : |z|\le 1\}$ , is it true that any element in $C(X , \mathbb C)$ can be uniformly approximated by polynomials , in $z, \bar z$ , with real co-efficients ? If we wanted ...
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0answers
33 views

If a sequence of polynomials converges uniformly to a continuous function on the real line, then this function is a polynomial [duplicate]

I'm trying to prove that if $f\colon\mathbb{R}\to\mathbb{R}$ is continuous function and there is a sequence of polynomials $p_n$ that converges uniformly to $f$ on $\mathbb{R}$, then $f$ is a ...
2
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0answers
90 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\...
2
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0answers
181 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 ...
2
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0answers
82 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 $\epsilon>0$, ...
2
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1answer
589 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,...
1
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2answers
58 views

Question regarding Weierstrass theorem generalized to Stone-Weierstrass

Weierstrass theorem. Lef $f$ be a defined and continuous function in $[a,b]$. Given $\epsilon>0,$ there exists a polynomial $P$ such that $\vert f(x)-P(x)\vert<\epsilon,$ for all $x\in[a,b].$ ...
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2answers
457 views

Weierstrass Approximation Theorem on $\frac{1}{x}$

Good day, we have just covered uniform continuity and polynomial approximations of continuous functions and I do not quite think I have the hang of it. We were given an example but it wasn't ...
1
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1answer
65 views

polynomials converging point wise to $f$ on $\mathbb{R}$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be continuous. Would there exist a sequence of polynomials converging point wise to $f$ on $\mathbb{R}$? I know that it is true on a compact set in $\mathbb{R}...
1
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2answers
90 views

Span$\{1,\sin t,\sin ^2t,\cdots\}$ is dense in $\mathscr{C}[0,1].$

Let $\mathscr{A}$ be a vector space in $\mathscr{C}[0,1]$ generated by the functions $1,\sin t,\sin ^2t,\cdots.$ Show that $\mathscr{A}$ is dense in $\mathscr{C}[0,1]$. Theorem $($ Stone ...
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2answers
158 views

If $\int_0^1 f(x)x^{2n+1}dx=0$ for all but finitely many $n\in\mathbb{N}$, then $f=0$ on $[0,1]$. [closed]

Let $f$ be continuous real function. Assume $\int_0^1 f(x)x^{2n+1}dx=0$ for all but finitely many $n\in\mathbb{N}$, then $f=0$ on $[0,1]$. Use Stone-Weierstrass theorem (not change of variables.) *...
1
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3answers
92 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
34 views

Uniformly approximating $f\in \mathcal{C}([1,\infty))$ with polynomials where $\underset{x\rightarrow +\infty}{\lim} f(x)=a$

Suppose $f\in \mathcal{C}([1,\infty))$ and $\underset{x\rightarrow +\infty}{\lim} f(x)=a$. Show that $f$ can be uniformly approximated on $[1,\infty)$ by functions of the form $g(x)=p(1/x)$, where $p$ ...
1
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1answer
25 views

Polynomial Approximation gives zero function

I am asked to prove that for given $f:[0,1]\to \mathbb{R}$ continuous, if $\int_{0}^{1} x^{2n} f(x) dx=0$ for all $n\geq0$, then $f$ is the zero function. I managed to prove the statement when I ...
1
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1answer
1k views

Stone Weierstrass Theorem Proof (Rudin) change

Question: Suppose Rudin uses, $Q_n=c_n(1-x^4)^n$ instead of $Q_n=c_n(1-x^2)^n$, how would the proof of the Stone Weierstrass Theorem change? Here is Rudin's proof of the Stone Weierstrass Theorem: ...
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1answer
109 views

How do I use Weierstrass Approximation Theorem?

In the question below, I would have to solve for an upper estimate using Weierstrass Approximation Theorem, however I am not familiar with the theorem, how do I go about solving it? For any $\epsilon\...
1
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2answers
35 views

Proving existence and uniquennes of integral

Let $\mathcal{C}_0[a,b]$ be the set of continous real valued functions in $[a,b]$, and $\mathcal{P}_0[a,b]$ the subspace of polynomials. Admitting Weierstrass Theorem, show that there exists an unique ...
1
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1answer
307 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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1answer
55 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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1answer
33 views

Weierstrass Approximation Theorem Problem [closed]

In the question below, I would have to solve for an upper estimate using Weierstrass Approximation Theorem, however I am not familiar with the theorem, how do I go about solving it? For any $\epsilon\...
1
vote
2answers
110 views

Self-adjoint Algebra made my ideas Vanish. How to get them back?

Let $K$ be a finite set and $\mathcal{A}$ a family of functions on $K$ that is a self-adjoint algebra, separates points and vanishes nowhere. Prove that $\mathcal{A}$ must then already contain every ...
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0answers
32 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_{...
1
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1answer
89 views

Weierstrass's approximation theorem with polynomials $p_n$ of degree $n$ for all $n.$

By Weierstrass's approximation theorem, every continuous function $f$ supported on an compact interval can be uniform approximated by polynomials. But, is it true that for every continuous $f$ on $[0,...
1
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0answers
65 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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1answer
180 views

Stone-Weierstrass complex theorem

I have already proven the real version of this theorem: If $X$ is a compact metric space. Let $\mathcal{C}(X) := \left(\mathcal{C}(X, \mathbb{R}),\Vert.\Vert_{\infty} = \Vert .\Vert\right)$ $\...
1
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0answers
280 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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0answers
476 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 ...
1
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1answer
63 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)=\...
0
votes
1answer
30 views

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 ...
0
votes
2answers
47 views

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 ...
0
votes
1answer
124 views

$\int_0^1 t^{n\lambda}f(t)dt=0$ 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 $x\in[0,1]$. ...
0
votes
2answers
87 views

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?
0
votes
2answers
55 views

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 ...
0
votes
2answers
84 views

Approximation of continuous function by linear combination of exponentials

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 $...