# Tagged Questions

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### decreasing rearrangements: why the asymmetry of measure-preserving maps?

Ryff proved in 1970 that the decreasing rearrangement $f^*$ of a, say, continuous function $f:[0,1]\to\mathbb{R}$ admits a measure preserving map $\phi$ such that $f=f^*\circ\phi$. In general it is ...
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### On the convergence of the the function series $\sum_{n=0}^\infty(-1)^n\frac{f^{(n)}(x)}{n!}x^n$

Let $f$ be a smooth real function defined around origin. If we informally differentiate from the series $\hat{f}(x):=\sum_{n=0}^\infty(-1)^n\frac{f^{(n)}(x)}{n!}x^n$ term by term we get ...
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### Weierstrass Approximation Theorem with an additional condition

Good afternoon: Let $f$ be a continuous function defined on an closed interval $[a, b]\subset\mathbb R$. By Weierstrass Approximation Theorem, for any $\epsilon>0$, there is a polynomial $p$ such ...
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### Riemann-Lebesgue lemma for measures

Riemann Lebesgue Lemma states that Fourier transform of an $L^1$ function, $\hat{f}(\lambda)$ is continuous and goes to zero as $|\lambda|\to \infty$. If $\mu$ is a finite nonatomic measure then is it ...
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### For which metric measure spaces is the Hardy-Littlewood maximal operator not of weak type (1,1)?

Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. I'll denote the Hardy-Littlewood maximal operator - either centred or uncentred, I don't mind ...
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### Poincaré lemma in infinite dimensions

Hi everyone, Is the PoincarĂ© lemma true in infinite dimensions? Here's a precise statement: Let $X$ be a Banach (or maybe Hilbert) vector space, $U$ a simply connected open set in $X$. Is it true ...
I have a function $f:\mathbb{R}^+ \to \mathbb{R}^+$ that I want to prove is eventually concave - i.e. that there exists $\gamma _0 > 0$ such that for every $\gamma>\gamma_0$, $f(\gamma)$ is ...