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.

70 questions
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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$\int_0^1 t^{n\lambda}f(t)dt=0$ 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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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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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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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 ...
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 $C[0,1]$. ...
$X:=\{z \in \mathbb C : |z|\le 1\}$ ; 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 ...