# 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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### 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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### 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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### A density proof not via Stone-Weierstrass

Context: I have proved Weierstrass' theorem (polynomials are dense in $C[a,b]$) in two ways: one using Bernstein polynomials, and one using convolutions. You can also use Stone-Weierstrass theorem, ...
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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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### 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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### 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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### $\int_0^1 t^{n\lambda}f(t)dt=0$ 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 $x\in[0,1]$. ...
2answers
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### If $\int_0^1 f(x)x^{2n+1}dx=0$ for all but finitely many $n\in\mathbb{N}$, then $f=0$ on $[0,1]$. [closed]

Let $f$ be continuous real function. Assume $\int_0^1 f(x)x^{2n+1}dx=0$ for all but finitely many $n\in\mathbb{N}$, then $f=0$ on $[0,1]$. Use Stone-Weierstrass theorem (not change of variables.) *...
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