# 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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### Hardy-Littlewood-Sobolev for “componentwise product” of Riesz kernels

Let $d\in\mathbb N$ and $0<\alpha<d$. Define the Riesz kernel $K_\alpha(x):=|x|^{\alpha-d}$, and the associated convolution operator $$K_\alpha f(x):=\int\frac{f(y)}{|x-y|^{d-\alpha}}~dy.$$ The ...
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### A Bessel-like integral

I encounter the following integral when trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, $0\le\lambda\le1$, $p\ge0$, $q\ge0$ are ...
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### Equivalence of antiderivative in L1 sense and in the usual sense

We say that$\ f$ is differentiable w.r.t to $L_1$ if there exists a$\ g$ such that: $$\lim_{h\to 0}\left\Vert\frac{f(x+h)-f(x)}{h} - g(x)\right\Vert_1 = 0$$ where $\Vert \cdot \Vert_1$ is the $L_1$ ...
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### Approximation of Lipschitz functions

Let $X$ be an open set in $\mathbb{R}^N$ and call $C^{1,1}(X)$ the set of functions $X\rightarrow \mathbb{R}$ that are $C^1$ with Lipschitz first derivatives. I have a $C^{1,1}$ function $f$ and a ...
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### A function fitting method: can we recover binary step functions?

A function fitting method Data $(p_i,a_i) \in \mathbb{T}^m\times\{-1,1\}$, $i = 1,2,...n$. Let $C_{\lambda}(f) = \sum\limits_{i=1}^{n}(f(p_i)-a_i)^2 + \|f\|_{L^2}^2 + \lambda\|\nabla^kf\|_{L^2}^2$ ...
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### Aleksandrov maximum principle for semi-convex function

Definition. Let $u:\Omega \rightarrow \mathbb{R}$. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$. Note. Saying that $u$ is semiconvex is ...
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### Are periodic real-analytic functions dense in the Frechet space $C^\infty(S^1)$?

The question is as above: Are periodic real-analytic functions dense in the Frechet space $C^\infty(S^1)$? Here I gave $C^\infty(S^1)$ the metric topology in which the convergence is the uniform ...