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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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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2answers
49 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 ...
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1answer
29 views
Numerical methods for solving the inverse Weierstrass transform
I have a measurement technique that results in a frequency distribution of the results. Let's say x is the parameter of interest, the method gives you a distribution F(x). I have the idea that the ...
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1answer
18 views
Uniform convergence of a sequence of polynomials to a specified function
Suppose we have $f: [1, \infty) \to \mathbb{C}$ is continuous and $\lim_{x\to\infty}f(x)$ exists. Is it true that there exists a sequence of polynomials $p_n$ such that $p_n(\frac{1}{x}) \to f(x)$ ...
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1answer
32 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\...
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1answer
38 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 ...
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0answers
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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2answers
41 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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1answer
88 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
vote
1answer
75 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,...
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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
75 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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2answers
64 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 $...
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1answer
32 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 $p$ ...
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2answers
105 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
58 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
75 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 $\epsilon > 0$
...
2
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1answer
330 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 ...
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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 ...
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1answer
64 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 ...
0
votes
1answer
58 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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0answers
163 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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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 ...
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0answers
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 ...
2
votes
0answers
74 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 $\epsilon>0$, ...
0
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1answer
129 views
Vector-valued Weierstrass theorem
I'm looking for a version of Weierstrass's approximation theorem that works for a continuous function $f:D \to \mathbb{R}^d$.
Versions that I know of
Multivariate Weierstrass theorem? uses a ...
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2answers
432 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
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 ...
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1answer
130 views
Rudin proof change, 7.27.
In Rudin, we are given this corollary, 7.27 to the Stone Weierstrass Theorem:
where Thm 7.26 is the Stone Weierstrass Theorem:
Say instead I replaced $|x|$ in the corollary with a continuous ...
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1answer
984 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
173 views
Stone-Weierstrass complex theorem
I have already proven the real version of this theorem:
If $X$ is a compact metric space. Let $\mathcal{C}(X) :=
\left(\mathcal{C}(X, \mathbb{R}),\Vert.\Vert_{\infty} = \Vert
.\Vert\right)$ $\...
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1answer
57 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}...
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1answer
17 views
How to show $B_n(f)^2\le B_n(f^2)$
Show that $B_n(f)^2 \le B_n(f^2)$ where $B_n(f)$ is Bernstein's polynomials. HINT : Expand $B_n((f-a)^2)$
I really do not know how to solve it only a hint.
according to HINT
$$B_n((f-a)^2)=B_n(f^2-...
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0answers
291 views
Is the set of all polynomials dense in the set of all continuous functions that maps from a compact set to reals
Let $C$ be a set of all continuous functions that map from a compact set to the reals. Is the set of all polynomials dense in $C$, with respect to the sup metric?
Can I show that by using the Stone-...
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0answers
348 views
Proof of inequality in Weierstrass approximation theorem proof through probability
I found this exercise in my probability theory book. The problem guides you through a proof of the Weierstrass Approximation Theorem through probability theory. My question is only about part b, so ...
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1answer
23 views
Question on a proof that involves continuity
I was reading the proof here but I have a question on it.
It uses the claim that if the integral of $f^2$ over [0,1] is 0 then $f$ is 0 by continuity. I'm not sure how to show that using continuity. ...
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votes
1answer
343 views
Proving Peano's Existence Theorem by approximating with $C^{\infty}$ functions using Weierstrass' Theorem.
Let $f:B_r(x_0)\to\mathbb{R}^n$ be continuous. Prove there always exists a local solution $x:[0,\delta)\to\mathbb{R}^n$ satisfying
$$x(0)=x_0, \hspace{1cm} x'(t)=f(x(t)), \quad \forall t \in (0,\...
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1answer
137 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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2answers
87 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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1answer
155 views
Prove Weierstrass approximation theorem for 0.2(3)
I'm studying calculus, and recently I've found a task I cannot solve: prove Weierstrass approximation theorem for a sequence and find its limit. The sequence is:
...
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1answer
80 views
Prove $\exists$ a sequence of polynomials which converges to $f$ on any compact subset of $\mathbb{R}$. [duplicate]
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 $\mathbb{R}$.
Question: Since $K$ is ...
3
votes
1answer
170 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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2answers
153 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.)
*...
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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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votes
1answer
122 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 $x\in[0,1]$.
...
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2answers
85 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?
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0answers
111 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 ...
0
votes
2answers
140 views
Use Weierstrass Approximation to prove that $f(x)=0$. With questions on the detail.
Let $f$ be continuous on $[0,1]$ and suppose $\int_0^1f(x)x^ndx=0$ for all $n\in\mathbb{N}_0$. Prove that $f=0$.
*Let $\epsilon\geq 0$. By Weierstrass theorem, we can find a polynomial $p(x)$ such ...
0
votes
0answers
144 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 $C[0,1]$.
...
2
votes
1answer
39 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 ...
