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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45 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 ...
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54 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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38 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
42 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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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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43 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
92 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\...
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1answer
81 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 ...
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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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71 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
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 $p$ ...
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106 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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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
78 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 $\epsilon > 0$ ...
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1answer
333 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
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 ...
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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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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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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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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

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
140 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
439 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 ...
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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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132 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
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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
175 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
61 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
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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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314 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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361 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
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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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1answer
365 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
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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89 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
157 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
81 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 ...
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1answer
178 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
157 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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1answer
123 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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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?
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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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2answers
143 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 ...
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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 $C[0,1]$. ...
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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 ...