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Choosing properly the Gaussian (e.g. $e^{-π\vert x\vert^2}$) and the normalization for the Fourier transform, your equation becomes $$ \hat f(\xi)=e^{-π\vert \xi\vert^2}\hat \mu(\xi). $$ This implies …

$f$ is a continuous function thus can be considered as a distribution. For $\phi\in C^1_c(0,1)$, we have $$ \langle f',\phi\rangle=-\int_{\mathbb R} f(x) \phi'(x) dx=\lim_{\epsilon \rightarrow 0} \int …

Take $\Psi$ as the standard Cantor function: $\Psi(x)=0$ for $x\le 0$, $\Psi(x)=1$ for $x\ge 1$, continuous, nondecreasing, constant on each connected component of the complement of the Cantor ternar …

Let $(X,\mathcal M, \mu)$ be a measure space, where $\mu$ is a positive measure and $X$ is topological space. Let $\mathcal B$ the Borel $\sigma$-algebra on $X$. The measure $\mu$ is called a Borel m …

An open subset $E$ of $\mathbb R^{N}$ such that $\mathbf 1_E$ belongs to $BV$ is said to be with "finite perimeter" and this implies that $$ \mathcal H^{N-1}(\partial E)<+\infty, $$ where $\mathcal H …

Let $(X,\mathcal M, \mu)$ be a measured space where $\mu$ is a positive measure. Let $\lambda$ be a complex measure on $(X,\mathcal M)$. When $\mu$ is sigma-finite, the Radon-Nikodym theorem provides …

We have with $\chi\in C^\infty_c(\mathbb R)$, equal to $1$ near 0, $$ \frac{1}{\sinh t}=\frac{\chi(t)}{\sinh t}+\frac{1-\chi(t)}{\sinh t}. $$ The second function belongs to $L^1$ and Young's inequalit …

Let $h:\mathbb R\rightarrow \mathbb R$ continuous such that the second distribution derivative $h''$ is continuous and bounded. We define $$ \rho(x)=\vert{h(x)}\vert^{\frac{1}{2}}+\vert{h'(x)}\vert, $ …

No. You may ask the same equivalent question for a function in $L^2$. A function $u$ belongs to $L^2$ means measurability and $$ \int \vert u(x) \vert^2 dx<+\infty, $$ which implies $ \lim_{R\rightarr …

A simple-minded answer. The push forward of a measure is a triviality: take a measure space $(X,\mathcal M, \mu)$ and a mapping $f:X\rightarrow Y$. Then defining $\mathcal N=${$B\subset Y, f^{-1}(B)\ …

Let $(X,\mathcal M)$ be a measurable space and let $\lambda$ be a complex measure on $(X,\mathcal M)$. The total variation $\vert \lambda\vert$ is a positive measure on $(X,\mathcal M)$ with a finite …

Let $X=\sum_ja_j(x)\frac{\partial }{\partial x_j}$ be a $BV$ vector field in an open subset of $\mathbb R^n.$ Alberti's theorem says that $$ DX_s=(\frac{\partial a_j }{\partial x_k})_{1\le j,k\le n}= …

Let $f$ be in $L^1(\mathbb R^n)$, then $\hat f$ belongs to $L^\infty(\mathbb R^n)$ and is (uniformly) continuous with $\lim_{\vert \xi\vert\rightarrow +\infty} \hat f(\xi)=0$: this is the Riemann-Leb …

Let $H=\mathbf 1_{\mathbb R^+}$ be the Heaviside function and let us define $g$ by $ g(x)=H(x) x^{-1/2}. $ We note that $g$ is homogeneous with degree $-1/2$ and thus its Fourier transform is homogene …