Questions tagged [weierstrass-approximation]
For questions about or using the Weierstrass approximation theorem (or the Stone-Weierstrass theorem). The Weierstrass theorem states that if $f:[a,b]\to\mathbb R$ is continuous and if $\epsilon>0$, then there exists a polynomial $p$ 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.
150
questions
1
vote
0
answers
45
views
A following up question on Weierstrass function approximation
This is a following-up question of
Proof — Weierstrass Approximation Theory for derivatives
and $f \in C^{\infty}$
Basically I wanted to explore whether there exists a sequence of polynomials $p_n$ ...
- 573
0
votes
0
answers
6
views
Is $\mathcal{O}=\{\pi_{v,w}| \ (\pi,H)\in Rep_f^s(G),\ v,w\in H\}$ dense in $C(G)$? where $G$ is compact, $T_2$ group
Let $Rep_f^s(G)$ be the set of all finite dimensional, strongly continuous hilbert space representations of compact, $T_2$ group $G$. For $(\pi,H)\in Rep_f^s(G)$ and $v,w\in H$ define $\pi_{v,w}(g):=\...
- 2,365
0
votes
0
answers
24
views
Generalization of Weierstrass approximation theorem?
For approximating realistic structures that mimic fractal, no-where differentiable manifolds, I'd like to say that such continuous boundaries in $\mathbb{R}^3$ can be approximated by a series of ...
- 425
1
vote
0
answers
43
views
Proving the Weierstrass Approximation Theorem using Polygonal functions
This is exercise 6.7.8. from Abbott's Understanding analysis 2nd ed.
As the question title says, this section is about the Weierstrass Approximation Theorem (WAT). Abbott guides us to prove WAT ...
- 532
0
votes
0
answers
26
views
Uniform approximation of a continuous function on $[0,1]$ by polynomials with a control on the uniform norm of the polynomials
Let $f \in C([0,1])$. By the Weierstrass approximation theorem, it is possible to uniformly approximate $f$ on $[0,1]$ by a sequence of polynomials $P_i$.
(i) Can we also require that $\|P_i\|_\infty \...
- 795
8
votes
2
answers
150
views
Let f be a continuous real valued function on the compact interval [a,b]. Given ϵ>0, show that there is a polynomial p such that: |p(x)−f(x)|<ϵ
Let $f$ be a continuous real valued function on the compact interval $[a,b]$. Given $\epsilon > 0$, show that there is a polynomial $p$ such that:
$p(a)=f(a)$,
$p'(a)=0$
and $|p(x) - f(x)| < \...
1
vote
0
answers
58
views
Generalizing how well an arbitrary set approximates a number?
Suppose $A\subseteq\mathbb{R}$ and $x\in \mathbb{R}$. I want a measure $\mu(x,A)$ that determines "how well" $x$ can be approximated by $A$?
For example, if $A=\mathbb{Q}$, we use the ...
user1027188
4
votes
1
answer
85
views
Externalizing A Concrete Application of Double Negation Toposes
I'm trying to come up with concrete problems which can be solved via topos theory, and I've found a good case study which has been really instructive. I've spent the past few weeks trying to ...
- 25.6k
1
vote
1
answer
53
views
Convolution with Landau kernel $L_k$ is polynomial of degree at most k
Let $B=B_\frac{1}{2}(0) \subset \mathbb{R}^n$. I want to show that given a function $F \in C_c(\mathbb{R}^n)$, with supp $F \subset B$ that the convolution $F \ast L_k$ restricted on $B$ is a ...
4
votes
0
answers
56
views
$[a,b]$ be a compact interval in $\Bbb R$, $f,g\in C[a,b]$, $f=g$ iff. $\int_a^b x^n f(x)dx=\int_a^b x^ng(x)dx$ for all $n$ - proof assistance
$\newcommand{\d}{\,\mathrm{d}}\newcommand{\c}{\mathcal{C}}$I need to show the following:
Let $[a,b]$ be a compact interval in $\Bbb R$; for any two $f,g\in\c[a,b]$, $f=g$ on $[a,b]$ if and only if: $$...
- 8,144
2
votes
0
answers
65
views
Stone–Weierstrass approximation theorem for multivariable case.
Stone–Weierstrass approximation theorem says
"Let $A$ be a (complex) unital sub-algebra of $C(K, \mathbb{C})$, such that if $ f\in A$, then $\overline{f} \in A$, and $A$ separates points of $K$. ...
0
votes
0
answers
55
views
What are the applications of the second part of the Stone-Weierstrass theorem?
(Stone-Weierstrass) Let $K$ be compact HAusdorff and $A\subset C_\mathbb{R}(K)$ a (sub)algebra which sepearates points. Then either $\overline{A}=C_\mathbb{R}(K)$ or $\exists x_0$ such that $\overline{...
0
votes
0
answers
41
views
Is this conclusion on stone-weierstrass approximation true?
Prove or disprove that it's possible to approximate
the functions in $C([a,b],\mathbb R) $ uniformly with arbitrary error $\epsilon$ by functions in the
algebra $\mathcal A$ generated on $[a, b]$ by ...
- 617
3
votes
1
answer
156
views
How to prove Algebra generated by$\{1,e^{ix}\}$ is dense in $C([0,\pi],\mathbb C)$?
Is it possible to approximate every function $f\in C([0,\pi],\mathbb C)$ uniformly with arbitary precision bby functions in the algebra generated by $\{1,e^{ix}\}$?
I know it's true for $\{1,e^{ix},e^{...
- 617
-1
votes
1
answer
51
views
Analytic Continuation of a function defined in integral
Let $C$ be the boundary of a unit circle. The function $f:\mathbb{C} \backslash C \to \mathbb{C}$ is defined by $f(z)=1-\frac{1}{2\pi i} \int_C \frac{1}{\zeta-z} d\zeta$.
I found that $f(z)=0$ when $z$...
- 21
2
votes
1
answer
69
views
A $T_{3\frac{1}{2}}$ space is compact if Stone-Weierstrass theorem holds.
A $T_{3\frac{1}{2}}$ space is compact if Stone-Weierstrass theorem holds. One proof of this is illustrated in Hewitt's "Certain generalizations of the Weierstrass approximation theorem". A ...
- 974
1
vote
0
answers
51
views
Proof of Stone's generalization of Weierstrass theorem in PMA by Walter Rudin
I was going through a proof of Stone's generalization of Weierstrass theorem in the book PMA by Walter Rudin. The theorem is stated as such:
Theorem 7.32 Let $\mathscr{A}$ be an algebra of real ...
0
votes
0
answers
69
views
Baby Rudin Theorem 7.33
This is Baby Rudin Theorem 7.33:
As far as I understand in both of those cases: $f=\lambda g$ and $f=u+iv$, $u(x)>0$.
I don't understand why $u(x)>0$.
- 480
1
vote
0
answers
48
views
Approximating $\bar{z}$ on unit arc
Problem statement: Given a proper closed arc of the unit circle, is there a sequence of polynomials that approximates $\bar{z}$ on the arc?
My thoughts:
Well $\bar{z} = 1/z$ on the arc...But the ...
- 369
1
vote
0
answers
91
views
Baby Rudin Theorem 8.15: Approximate periodic continuous function by trigonometric polynomials
8.15 Theorem If $f$ is continuous (with period $2\pi$) and if $\epsilon>0$, then there is a trigonometric polynomial $P$ such that
$$\left|P(x)-f(x)\right|<\epsilon$$
for all real $x$.
Proof ...
- 674
0
votes
1
answer
131
views
Weierstrass approximation theorem in two dimensions
Show that polynomials $P(x,y)=p(x)q(y)$ are dense in $C([0,1]^2,\mathbb{R})$ (i.e. the set of continous functions in two variables).
My attempt:
Based on inspiration from Multivariate Weierstrass ...
- 159
2
votes
2
answers
82
views
$C_0^\infty(\mathbb{R}^k)$ functions can be approximated via polynomials
Let $f:\mathbb{R}^k\rightarrow \mathbb{R}$ be a vanishing at infinity function, also infinitely differentiable, i.e. $f\in C_0^\infty(\mathbb{R}^k,\mathbb{R})$. Is it true that I can always ...
2
votes
2
answers
109
views
$\int_0^1 f(x)e^{nx} \, dx=0$ for all $n \in \Bbb N\cup{\{0\}}$ implies $f(x) = 0$ [closed]
$f$ is a continuous real valued function $f: [0,1] \to\mathbb R$ and
$\int_0^1 f(x)e^{nx} \, dx=0$ $\forall n \in N\cup{\{0\}} \implies f(x) = 0$ on the interval $[0,1]$
I am trying to prove this ...
- 341
2
votes
0
answers
32
views
Can polynomials with coefficients in [-1, 1] uniformly approximate arbitrarily large constant functions?
Consider the set of polynomial functions $S := \{p \in C[2, 3]: p$ is a non-zero polynomial function on $[2, 3]$ with coefficients in $[-1, 1]$$\}$, is it true that $\inf_{f \in S} \{ \omega(f) + (\...
- 529
2
votes
2
answers
126
views
Proof — Weierstrass Approximation Theory for derivatives
I'm working through the second edition of Abbott's Understanding Analysis, and I'm stuck on the following (6.7.11):
Assume that f has a continuous derivative on $[a, b]$. Show that there exists a ...
- 21
0
votes
0
answers
16
views
Relationship between the multiplication of two real functions and their approximation polynomials (Weierstrass approximation theorem)
Consider two continuous real valued functions $f$, and $g$ over the interval $[a,b]$. For a given $\epsilon > 0$, for each function the we can use the Weierstrass approximation theorem to express ...
- 273
1
vote
1
answer
54
views
Relationship between the composition of two real functions and their approximation polynomials (Weierstrass approximation theorem)
Consider two continuous real valued functions $f$, and $g$ over the interval $[a,b]$. For a given $\epsilon > 0$, for each function the we can use the Weierstrass approximation theorem to express ...
- 273
1
vote
2
answers
44
views
Approximation of continuous functions vanishing at a point with polynomials vanishing at the same point
Let $C[0, 1]$ denote the set of all real-valued continuous functions on $[0, 1]$.
Consider the normed linear space
$$
X=\{f\in C[0,1]| f(\frac{1}{2})=0\}
$$
with the sup-norm
$$||f|| = \sup\{|f(t)| : ...
- 187
0
votes
0
answers
59
views
Use of Stone-Weierstrass Theorem for $1/z$ on complex circle
Let $C:=\{x\in\mathbb{C}:|x|=1\}$ be the complex circle and $f:C\rightarrow\mathbb{C}$ the mapping $x\mapsto 1/x$.
From the Stone-Weierstrass theorem I get $\oint fdx =0$, but by Cauchy's theorem $\...
- 860
0
votes
0
answers
32
views
Sequences of integrals involving derivatives of approximating functions
Let $f, g: [0,1] \to \mathbb{R}$ be continuous functions. By the Weierstrass approximation theorem, we can find sequences $(f_n)_{n\in\mathbb{N}}$, $(g_n)_{n\in\mathbb{N}}$ of smooth functions, in ...
- 1
0
votes
0
answers
19
views
How to prove that representable functions are dense in the space of Lipschitz functions.
Given a separable compact metric space $(E,\mathrm{d})$ for any $x\in E$ we call $\pi_x\in\mathrm{Lip}(E)$ the Lipschitz function represented by $x$ and defined by
$$
\pi_x(y)=\mathrm{d}(x,y).
$$
We ...
- 1,420
1
vote
1
answer
158
views
Using weierstrass approximation theorem prove that polynomials are dense in $C(X,\Bbb{R})$.
Weierstrass approximation theorem says-
Set of all polynomials on a closed interval $[a,b]$ is dense in $C[a,b]$.
Using this result, we have to prove that Set of all polynomials on a closed and ...
- 327
2
votes
2
answers
171
views
Does there exist a sequence of polynomials converging uniformly to $f(x) = |x| $ on $ℝ$?
Does there exist a sequence of polynomials converging uniformly to $f(x) = |x|$ on $ℝ$?
I understand this is possible on a closed interval due to Weierstrass Approximation Theorem.
Intuitively, I ...
- 301
1
vote
0
answers
48
views
For each $\epsilon>0$ there is a polynomial $p\colon\mathbb{C}\to\mathbb{C}$ such that $|p(x)-|x||<\epsilon$ for all $x\in[-r,r]$.
I'm looking for an elementary proof (e.g. no Stone-Weierstrass) of the following result:
Fix an $r\geq0$. Then for each $\epsilon>0$ there exists a polynomial $p\colon\mathbb{C}\to\mathbb{C}$ such ...
- 2,958
2
votes
1
answer
62
views
An application of Weierstrass Approximation Theorem to polynomial
The question is
Let $f:[0,1]\to\mathbb{R}$ be a continuous function. Let $\varepsilon>0$, show that there is a polynomial $p:[0,1]\to\mathbb{R}$ with $\vert f(x)-p(x)\vert <\varepsilon$ for all ...
- 129
0
votes
2
answers
43
views
Value of sum $\sum_{k=0}^{n}n_{C_k}\frac{k}{n}x^k(1-x)^{n-k}$ .
$\sum_{k=0}^{n}n_{C_k}\frac{k}{n}x^k(1-x)^{n-k}$ is equal to
$1.$ $1$.
$2.$ $x$.
$3.$ $x^2$
$4.$ $x^n$.
According to me $B_n(x)=\sum_{k=0}^{n}n_{C_k}\frac{k}{n}x^k(1-x)^k$ form polynomials called ...
- 5,268
1
vote
0
answers
53
views
Show $f$ is the uniform limit of $p_n\circ \cos,$ where the $p_n$ are polynomials.
If $f : [−\pi, \pi]$ is continuous and even on $[−\pi, \pi],$ show there exists a polynomial sequence $\{p_n\}$ such that $p_n\circ\cos$ converges uniformly to $f$ in $[−\pi, \pi]$
I was trying to ...
- 11
0
votes
1
answer
212
views
How to prove $f$ can be uniformly approximated on $[1,\infty )$ by a function $g$, using Weierstrass theorem. [duplicate]
Let $f:[1,\infty )\to \Bbb R$ be continuous, and suppose that $\lim \limits_{x \to \infty}f(x)$ exists. Use the Weierstrass approximation theorem to prove that $f$ can be uniformly approximated on $[1,...
- 1,019
-1
votes
1
answer
43
views
Prove Weierstrass theorem for a general $[a,b]$ follows from the result on $[0,1]$. [closed]
Give a detailed proof of the assertion that the Weierstrass theorem for a general $[a,b]$ follows from the result on $[0,1]$ (by using lemma 11.1)
It's been two days since I am trying to figure out ...
1
vote
1
answer
71
views
What does the phrase "by a polynomial in a finite number of functions of $\mathcal{F}$ means?
In Royden's "Real Analysis" (second edition), there is the following exercise concerning the Stone-Weierstrass Theorem (it can be found on page 175, Chap. 9, Sec. 7):
33. Let $\mathcal{F}$ ...
- 38
0
votes
1
answer
40
views
Inequality of expectation in Weierstrass theorem
The proof I was given in class for Weierstrass Theorem using Berstein Polynomials contains this one particular inequality:
$|f(x) − E[f(\frac{S_n}{n})]| ≤ E[|f(x) − f(\frac{S_n}{n})|]$
E stands for ...
0
votes
2
answers
39
views
If $p$ is a polynomial approximating a continuous function $h$, why can we assume $\left\|p\right\|_\infty\le\left\|h\right\|_\infty$?
Let $K\subseteq\mathbb R$ be compact and $h:K\to\mathbb R$ be continuous and $\varepsilon>0$. By the Stone-Weierstrass theorem, there is a polynomial $p:K\to\mathbb R$ with $\left\|h-p\right\|_\...
- 12.8k
2
votes
0
answers
56
views
Rate of approximation by polynomials to a C1 function(Weierstras Approximation thm)
By Weierstrass Approximation theorem, a continuous function on the interval $[0,1]$ can be uniformly approximated by polynomials. But if the function $f$ is continuously differentiable on $[0,1]$, ...
- 651
2
votes
2
answers
296
views
Using Stone-Weierstrass theorem prove that $f\equiv 0$ in $[0,1]$ under certain conditions.
Let $f$ be a continuous function over $[0,1]$ and $\displaystyle \int_0^1 f(x)\,dx=0$. If for any positive integer $n$, $\displaystyle \int_0^1 x^{12+3n}f(x)\,dx=0$ then using Stone-Weiersterass ...
- 12.3k
0
votes
1
answer
58
views
Riemann-integral and Stone-Weierstrauss Theorem
Question:
Let ƒ be a continuous real-valued function on [0, 1].
Show that
$\lim_{n->\infty }\frac{\int_{0}^{1}x^nf(x)dx}{\int_{0}^{1}x^ndx} = f(1)$
My approach: I thought the denominator and the ...
- 321
0
votes
0
answers
189
views
$ \int_a^bx^nf(x)\mathrm{d}x=0,\ n=0,1,2,\cdots. $ Prove that $f=0$. [duplicate]
$f$ is a continuous function on $[a,b]$. If
$$
\int_a^bx^nf(x)\mathrm{d}x=0,\ n=0,1,2,\cdots.
$$
Prove that $f(x)=0$.
A further problem is we only have
$$
\int_a^bx^nf(x)\mathrm{d}x=0,\ n=1,2,\cdots.
$...
- 1,123
0
votes
1
answer
40
views
$\int_0^1 f(t)e^{nt} dt = 0 \implies f \equiv 0 $ an application of Stone-Weierstrass
In one of the questions on my problem sheet, I've been asked to prove that if there is an $f \in C[0,1]$ such that $ \int_0^1 f(t) e^{nt} dt = 0 $ for all $n \in \mathbb{N} $ then $ f \equiv 0 $.
...
0
votes
3
answers
87
views
A related question to the Weierstrass approximation theorem
Suppose $f : [a,b] \to \mathbb{R}$ is a continuous function. Then, Weierstrass Approximation Theorem asserts that there is a polynomial on $[a,b]$ which is as close to $f$ as we want (in a uniformly ...
3
votes
1
answer
74
views
Weierstrass Approximation generalization
By the standard Weierstrass theorem, if $f:[0,1]\to R$ is continuous then
$$\sum_{j=0}^n f(j/n)\binom{n}{j}x^j(1-x)^{n-j}$$
converges to $f(x)$ uniformly for $0\le x\le 1$. I am wondering if there was ...
- 644
0
votes
0
answers
23
views
Equivalent statement of Weierstrass approximation theorem.
Let $f:[a,b]\to \Bbb C$ be a continuous function on $[a,b]$.
Then there exists a sequence $\{p_n\}$ of polynomials such that $$p_n\to f \text{ uniformly on }[a,b] \;\;\;(1)$$
Equivalently, for all $\...
- 1,019