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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questions
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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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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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74 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
$$
\int_a^bx^nf(x)\mathrm{d}x=0,\ n=1,2,\cdots.
$...
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1answer
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$\int_0^1 f(t)e^{nt} dt = 0 \implies f \equiv 0 $ an application of Stone-Weierstrass
In one of the questions on my problem sheet, I've been asked to prove that if there is an $f \in C[0,1]$ such that $ \int_0^1 f(t) e^{nt} dt = 0 $ for all $n \in \mathbb{N} $ then $ f \equiv 0 $.
...
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3answers
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A related question to the Weierstrass approximation theorem
Suppose $f : [a,b] \to \mathbb{R}$ is a continuous function. Then, Weierstrass Approximation Theorem asserts that there is a polynomial on $[a,b]$ which is as close to $f$ as we want (in a uniformly ...
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Weierstrass Approximation generalization
By the standard Weierstrass theorem, if $f:[0,1]\to R$ is continuous then
$$\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=\...
1
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1answer
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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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1answer
21 views
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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1answer
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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1answer
115 views
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 ...
3
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1answer
53 views
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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0answers
76 views
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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3answers
44 views
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$$\mathbb{R}$ be a function (not necessarily continuous) which has a maximum and minimum on $\mathbb{R}$.Let $f:$ $\mathbb{R}$ be a function which is continuous on ...
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2answers
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Why is sequence $\sum_{k = 0}^{n}\frac{(-1)^{k}}{(2k + 1)!}x^{2k + 1}$ not a sequence of polynomials converging to $f(x) = \sin x$?
It is a fact that $\sin x = \sum_{k = 0}^{\infty}\frac{(-1)^{k}}{(2k + 1)!}x^{2k + 1}$. It would appear that $\sum_{k = 0}^{n}\frac{(-1)^{k}}{(2k + 1)!}x^{2k + 1}$ is a sequence of polynomials ...
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1answer
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Weierstrass-M test find $a_n$
We have $\sum_{n=2}^{\infty} \frac{log(n)^x}{n}$ for $x\in$R.
We let r<-1 and I have to show that $\sum_{n=2}^{\infty} \frac{log(n)^x}{n}$ is uniform convergent for all x $\leq$r. I think I can ...
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1answer
62 views
Weierstrass Approximation - Proof that Integral Exists
Trying to answer the following question;
Let $f(x)$ be a continuous real valued function on $[0,4]$. Given any $\epsilon>0$ prove there is a polynomial $p(x)$ such that $$\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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1answer
69 views
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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1answer
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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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2answers
52 views
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)|\...
1
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1answer
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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2answers
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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0answers
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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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1answer
49 views
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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61 views
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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4answers
65 views
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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1answer
75 views
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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1answer
67 views
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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0answers
41 views
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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0answers
39 views
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 ...
1
vote
2answers
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 ...
1
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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$ ...
1
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2answers
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 ...
1
vote
1answer
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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1answer
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.
$$
\...
0
votes
2answers
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 ...
0
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2answers
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 ...
0
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1answer
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 ...
