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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63 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 $\mathcal{F}$ ...
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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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Derive the Stone-Weierstrass on a general topological space from the compact space case and the Weierstrass approximation theorem

Let $E$ be a topological space; $K$ be a compact Hausdorff subset of $E$; $\mathcal C\subseteq C_b(E)$ be an algebra with $1\in\mathcal C$ such that $\{\left.f\right|_K:f\in\mathcal C\}$ is point-...
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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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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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40 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 ...
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Weierstrass approximation theorem for periodic functions

In one of my textbooks it says that Weierstrass approximation theorem states that every continuous periodic function $h(t)$ is the uniform limit of trigonometric polynomials. This then implies that $$...
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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 $\...
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$f\in C [-1,1]$ and $\int_{-1}^1 f(x) x^{2n} dx=0$ implies $f$ is odd function?

Let $f:[-1,1] \to \mathbb{R}$ be a continous function such that $\int_{-1}^1 f(x) x^{2n}dx=0$ for all $n\ge 0$ . Which of the following statements is necessarily false ? $(1) \int_{-1}^1 f(x)^2 dx=\...
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A statement of double integral which is closely related to Weierstrass approximation theorem

we know that due to the Weierstrass approximation theorem. if we have a nonnegative function $f:[a,b] \to \mathbb{R}$ and f is continuous and $\int_{a}^{b}f(x)dx=0$, then we will have $f=0$ on $[a,b].$...
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Prove any smooth function with compact support can be uniformly approximated by elements in the space generated by $f_a(x)=e^{iax}e^{-x^2}$.

Below is question 16 from chapter III of Lang's Real and Functional Analysis: For $a\in \Bbb R$ let $f_a(x)=e^{iax}e^{-x^2}$. Prove that any function $\varphi$ which is $C^\infty$ and has compact ...
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How to insert a polynomial between two continuous functions

Let $f,g\in C[0,1]$ such that $f(x)<g(x)$ for all $x\in [0,1]$. Show that there exists a polynomial $p\in C[0,1]$ such that $f(x)<p(x)<g(x)$ for all $x\in [0,1]$. I know that I have to ...
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39 views

Proof of Stone Weierstrass Theorem from Hahn Banach

It can be found here a proof of Stone-Weierstrass Theorem through Hahn-Banach theorem (hyperplane separation of convex sets). I find one line in the proof difficult to understand: Suppose, for the ...
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Prove that there a non-zero continuous function $f$ on $[-1,2]$ for which $\int_{-1}^2 x^{2n} f(x) \; dx = 0$ for all $n \geqslant 0$.

I've been trying to find an example of a function $f \in \mathcal{C}[-1,2]$ with $f \neq 0$ such that $\int_{-1}^2 x^{2n} f(x) \; dx = 0$ for all $n \geqslant 0$, but I'm finding it very difficult. I ...
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Weierstrass approximation polynomial with $p^{(i)}(0)=0$

Given a continuous real function $f:[0,1]\to\mathbb{R}$, for $k\in\mathbb{N}$, we need to find a rational polynomial $p$ satisfying $p^{(i)}(0)=0$ $(1\leq i\leq k-1)$ such that for $\epsilon>0$, $\...
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Using the weierstrass theorem to prove a solution to a minimization problem exists

Question: Use the Weierstrass Theorem to show that a solution exists to the expenditure minimization problem of subsection 2.3.2, as long as the utility function II is continuous on $\mathbb{R}$ and ...
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How do you prove a function g has a max and min and that f of g does not have a max on R?

Question: Let g: $\mathbb{R}$ $\to$$\mathbb{R}$ be a function (not necessarily continuous) which has a maximum and minimum on $\mathbb{R}$.Let $f:$ $\mathbb{R}$ be a function which is continuous on ...
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Why is sequence $\sum_{k = 0}^{n}\frac{(-1)^{k}}{(2k + 1)!}x^{2k + 1}$ not a sequence of polynomials converging to $f(x) = \sin x$?

It is a fact that $\sin x = \sum_{k = 0}^{\infty}\frac{(-1)^{k}}{(2k + 1)!}x^{2k + 1}$. It would appear that $\sum_{k = 0}^{n}\frac{(-1)^{k}}{(2k + 1)!}x^{2k + 1}$ is a sequence of polynomials ...
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Weierstrass-M test find $a_n$

We have $\sum_{n=2}^{\infty} \frac{log(n)^x}{n}$ for $x\in$R. We let r<-1 and I have to show that $\sum_{n=2}^{\infty} \frac{log(n)^x}{n}$ is uniform convergent for all x $\leq$r. I think I can ...
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Weierstrass Approximation - Proof that Integral Exists

Trying to answer the following question; Let $f(x)$ be a continuous real valued function on $[0,4]$. Given any $\epsilon>0$ prove there is a polynomial $p(x)$ such that $$\int_0^4|f(x)-p(x)|^2dx&...
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Uniform convergence - Is f continuous and is f differentiable $f(x) = \sum_{n=1}^{\infty}\frac{1}{2n^2-\sin(nx)}$

Let f(x) = $\sum_{n=1}^{\infty}$$\frac{1}{2n^2-\sin(nx)}$ ($x\in\mathbb{R}$) (a) Decide whether f is continuous on R. (b) Is f differentiable? Don't even know where to begin with this question, I ...
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Approximation to continuous functions over an closed interval

Let $$f\in C[a,b]$$ A triangular system is a series of numbers \begin{matrix} x_{11}\\ x_{21}&x_{22}\\ x_{31}&x_{32}&x_{33}\\ \cdots \end{matrix} that $$a<x_{n1}<x_{n2}<\cdots<...
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Proof that Fourier Series of a function is unique

I need to prove the following, which is allegedly the same as proving the uniqueness of a function's fourier series representation: Let $f \in C^{2\pi}$ such that $\int^{\pi}_{-\pi}f(x)cos(nx)dx=0$ ...
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Stone–Weierstrass theorem exercise

$f$ is continuous and $$ \int_{a}^{b} x^nf(x) dx=0 $$ for every $n\leqslant N$. Prove that $f$ has at least $N+1$ zero points at $(a,b)$.
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Can Weierstrass's theorem be specialized to symmetric functions and symmetric polynomials?

Weierstrass's theorem says that continuous functions can be uniformly approximated by polynomials. Can one have a similar theorem saying that symmetric functions can be uniformly approximated by ...
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Does integration preserve uniform convergence of sequence? (Weierstrass Approximation Theorem)

Trying to solve the following problem: Let $f(x)$ be a continuous real-valued function on $[0,3]$. Given any $\varepsilon>0$ prove there exists a polynomial, $p(x)$, such that $\int_0^3|f(x)-p(x)|\...
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74 views

Prove the uniform convergence of Fourier series

Suppose $f \in C^1$ with a period of $2\pi$. Its Fourier series: $$ f \sim \sum_{n=-\infty}^{\infty} \widehat{f}(n) \mathrm{e}^{\mathrm{i} n t} $$ I want to prove that: $$ \left\|f-S_{N}(f)\right\|_{\...
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63 views

Prove polynomial sequence converging to f exists (Weierstrass Approximation Theorem)

I have the following problem: Prove there is a sequence of polynomials, $p_n(x)$, such that $p_n(x)$ converges uniformly to $|x|$ on [-1,1] and $p_n(0)=0$ for all n. I'm not sure if I'm ...
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Weierestrass's theorem for manifolds?

In this post the answer states that "by Weierstrass' approximation theorem, Annie's question is equivalent to: when is $C^\infty(M,N')$ not $C^0$-dense in $C^\infty(M,N)$" However, what is ...
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approximation to the identity and polynomial

(a) Prove that for any real number $0 \le y \le 1$ and any natural number $n \ge 0$, that $(1-y)^n \ge 1-ny$. (b) Show that $\int_{-1}^1 (1-x^2)^n \ge \frac1{\sqrt{n}}$. (Hint: for $|x| \le 1/\sqrt{...
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Let $f:[-1,1] \rightarrow \mathbb{R}$ be a continuous function. If $\int_{-1}^{1} x^{2n}f(x) dx = 0$, what can you say about f?

Let $f:[-1,1] \rightarrow \mathbb{R}$ be a continuous function. If $\int_{-1}^{1} x^{2n}f(x) dx = 0$, for all $n = 0,1,2,..$ what can you say about f? I have seen a similar proof for $\int_{0}^{1} ...
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polynomials in $t^2$ is dense in $C([a, b])$ if $0 \notin [a,b]$

My instructor mentioned as an application of stone-weistrass that the set of polynomials in $t^2$ is dense in $C([a, b])$ if $0 \notin [a,b]$. I could not be able to prove the density to understand ...
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Separability of $C^2_b(\Omega)$

Let $\Omega \subseteq \mathbb R$ be open and bounded. Denote by $C_b^2(\Omega)$ the space of functions $f:\Omega\rightarrow \mathbb R$ such that $f, f', f''\in C_b(\Omega)$. Endow this space with the ...
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Uniform approximation of an analytic function and its derivatives

The Lavrentiev's theorem is stated as follows Let $K \subset \mathbb{C}$ be a compact set. Then every continuous function $f: K\to \mathbb{C}$ can be approximated uniformly by polynomials if and ...
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Application of Stone-Weierstrass (Proof verification)

Let $f:[0,1]\rightarrow \mathbb{R}$ be continuous.Assume $\int_{0}^{1}f(t)e^{-nt}dt=0$ whenever $n$ is a non-negative integer. Does it follow $f$ is identically $0$? Does the answer change if $-nt$ is ...
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What functions can be approximated by homogeneous rational functions with numerator and denominator of the same degree?

Consider homogeneous polynomials $P(X,Y)$ in two variables. According to Approximation by homogeneous polynomials by Totik, "every even continuous function on a centrally symmetric convex curve can be ...
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1answer
52 views

Find a sequence of functions ${p_n}$ which converges uniformly to $f$ on $[0,1]$ with $p_n(0)=p_n(1)=0.$

Suppose $f$ is a continuous function on $[0,1]$ such that $f$ is $0$ at both the endpoints and I have to prove that there is a sequence of polynomials coverging uniformly to $f$ and having the ...
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115 views

A sequence of polynomials converging pointwise to a given continuous function on $\mathbb R$ convergence being uniform in every compact interval.

How to prove that given any continuous function from $\mathbb R$ to $\mathbb R$ we can get a sequence of polynomials converging to that function pointwise and the convergence is uniform on any compact ...
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1answer
82 views

Applications Stone-Weierstrass Theorem

Problem: Let $X$ and $Y$ be compact spaces. Prove that for any real-valued function $f$ on $X \times Y$ and any $\epsilon > 0$ we can find continuous real-valued functions $g_1,g_2,g_3,\dots,g_n$ ...
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61 views

$f \in C([-1,1])$, given $\int_{-1}^{1}f(x)x^ndx = 0$ for $n = 0,1,2,…$ then $f(x)=0, \ \forall x \in [-1,1]$

Let $f:[-1,1] \to \mathbb{R}$ be continuous on $[-1,1]$. Assume $\displaystyle \int_{-1}^{1}f(x)x^ndx = 0$ for $n = 0,1,2,...$ Then show $f(x)=0, \ \forall x \in [-1,1]$ I would like to ...
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106 views

Stone-Weierstrass Theorem (Lattices)

I am struggling with a portion of a proof concerning the lattice version of the Stone-Weierstrass theorem. In particular, there is a subset $\mathcal{A}$ of the set of all real-valued continuous ...
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118 views

approximation of 'any' bounded continuous function using bounded continuous functions with compact support

Suppose that $\phi$ is a bounded, continuous function with compact support $I$ (i.e. a bounded interval), then given any $\epsilon > 0$, there exists a simple function $\phi_{\epsilon}$ s.t. $$ \...
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101 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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93 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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128 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 ...