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.

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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 $p_n$ ...
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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):=\...
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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 ...
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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 ...
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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 \...
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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)| < \...
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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 ...
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4 votes
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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 ...
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1 answer
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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 ...
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$[a,b]$ 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: $$...
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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 $K$. ...
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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{...
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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 ...
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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^{...
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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 $z$...
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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 ...
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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 ...
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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 $u(x)>0$.
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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 ...
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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   ...
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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 ...
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$C_0^\infty(\mathbb{R}^k)$ 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 ...
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2 votes
2 answers
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$\int_0^1 f(x)e^{nx} \, dx=0$ 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 ...
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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) + (\...
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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 ...
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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 ...
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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 ...
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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)| : ...
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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 $\...
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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 ...
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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 ...
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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 ...
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2 votes
2 answers
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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 ...
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1 vote
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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 ...
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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 ...
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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 ...
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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 ...
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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,...
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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 ...
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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 $\mathcal{F}$ ...
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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 ...
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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\|_\...
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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 $[0,1]$, ...
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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 ...
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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 ...
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$ \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. $...
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$\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 $. ...
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3 answers
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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 ...
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3 votes
1 answer
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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 ...
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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 $\...
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