# 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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### 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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### 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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### 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 ...