# Questions tagged [real-analysis]

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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### Hilbert transform on weighted Sobolev spaces

Let $\mathscr H\,f$ denote the Hilbert transform of a function $f \in L^2(\mathbb R)$. We know that $\mathscr H$ is an isometry on $L^2(\mathbb R)$, but I want to know to what is the mapping ...
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### Sufficient and necessary conditions for decomposing the sum of random variables

Given two $n$-tuple vectors $\vec{\alpha}=(\alpha_1,\cdots,\alpha_n)$ and $\vec{h}=(h_1,\cdots,h_n)$, where $h_i\ge0$, $\sum_{i=1}^nh_i=1$, and $\alpha_i\in(0,1)$, we consider a random variable $S$ on ...
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### To show a analytic map is zero from a property regarding logarithmic integral

Let $F$ be analytic on $\mathbb{H}=\{z\in\mathbb{C}:Im(z)>0\},$ continuous upto $\overline{\mathbb{H}}$ and bounded on each of the half plane $\{Im(z)\geq h>0\}.$ How to show that if $F$ ...
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### Convergence of a certain serie

I came cross the following serie : $$\sum\limits_{\mathbf k \in \mathbb N^d} e^{\langle \mathbf r, \mathbf k \rangle}$$ What would be the conditions on the d-dimensional real vector $\mathbf r$ for ...
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### Best bounds on the integral of an increasing function

The following question, somewhat edited here, was asked and then closed at The best bound of the integral of a nondecreasing real function in a closed interval. Let $F\colon[0,1]\to[0,1]$ be a ...
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### Another uniform estimation of an integral involving an Hölder function with derivative that is Hölder

Let $\Omega\subset\mathbb{R}^n$, let $s\in [1/2,1)$, let $u\in C^{1,2s-1+\epsilon}(\Omega)$ such that: $u=0$ on $\mathbb{R}^n\setminus\Omega$, and: $u\in C^{0,s}(\mathbb{R}^n)$, is true that there ...
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