# 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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### $\int_0^1 f(t)e^{nt} dt = 0 \implies f \equiv 0$ 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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### 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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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 \$\...