# Questions tagged [schwartz-distributions]

A distribution is a continuous linear functional on the space $\mathcal{C}^{\infty}_c$ of smooth (indefinitely differentiable) functions with compact support. Though they appeared in formal computations in the physics and engineering literature in the late $19^{th}$ century, their formal setting was brought up by the work of S. Sobolev and L. Schwartz in the middle of the $20^{th}$ century.

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### How to generalize the various vector calculus theorems to distributions?

Here is a list of vector calculus identities; in the proof of these identities, we all assume that these functions are $𝐶^𝑘$ in an open set, and we usually use these identities to calculate ...
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### “Potential” for a divergence-free distribution [duplicate]

Let $S(\mathbb R^n)$ denote the space of all Schwartz functions on $\mathbb R^n$ equipped with the topology induced by the usual Schwartz semi-norms. Let $S(\mathbb R^n)^*$ denote its dual. For a ...
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### Uniqueness of distributional solutions to the Poisson equation

Let $S(\mathbb R^n)$ denote the space of all Schwartz functions on $\mathbb R^n$ equipped with the topology induced by the usual Schwartz semi-norms. Let $S(\mathbb R^n)^*$ denote its dual. My ...
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### Derivation in Sobolev space [closed]

Let $f\in W^{1,\infty}(]0,T[)$ $(0<T\le\infty)$ such that $f(x)>0$ a.e. $x\in\mathopen]0,T[$ and let $$g(x)=e^{-\int_0^x \frac{ds}{f(s)}}$$ Formally $g' = -\frac{1}{f}g$. How can I justify this ...
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### How to understand subharmonic functions, distributions, and measure?

Sorry if this turns out to be a silly question, but I am having difficulties in both understanding it and finding other references for it. I hope that someone can clear my concepts here on overflow. ...
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### English translation of Schwartz's papers on vector-valued distributions

I am interested in systematically studying the theory of vector-valued distributions. The original two papers due to Laurent Schwartz entitled Théorie des distributions à valeurs vectorielles. I & ...
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### Literature on the product of two distributions satisfying the Hörmander condition

I am currently studying some basic questions concerning the product $uv\in \mathscr{D}'(\mathbb R^n)$ of two Schwartz distributions $u,v\in \mathscr{D}'(\mathbb R^n)$ satisfying the Hörmander ...
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### Poincaré's Lemma in the space of tempered distributions

It is well known that if $f\in \mathcal{D}'(\mathbb{R}^3,\mathbb{R}^3)$ and $\textbf{curl} f= 0$ then there exists a $u\in \mathcal{D}'(\mathbb{R}^3)$ such that $\nabla u = f$. Question. Does the ...
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### Fourier Transform ; half space elliptic baby problem

I am attempting to look at some Liouville type theorems via a Fourier analysis approach and after looking at a baby problem I seem to be very confused. I assume this doesn't count as a research ...
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It is well known that the class of Schwartz functions $\mathcal{S}$ in dense in all $L^p$ spaces therefore for each $f \in L^2$ there exists a sequence of Schwartz functions $(f_k)$ such that $\lVert ... 0answers 157 views ### Feynman path integral and Wilsonian renormalization Everything below is to be viewed in the Euclidean setting with$d$dimensions and all measures are understood to be Borel measures. The usual problem of Quantum Field Theory is to make sense of ... 0answers 77 views ### Support of a fundamental solution of wave equation The solution of the wave equation $$\Box E = \delta$$ is $$E(t,x) = \mathscr{F}^{-1} \left( \frac{\sin (t\lvert \cdot \rvert ) }{\lvert \cdot \rvert} \theta (t) \right)(x)\in\mathcal{S'}(\mathbb{R^{... 0answers 42 views ### Functions on dense subgroups of \mathbb{R}^n Let G be a finitely generated dense subgroup of \mathbb{R}^n, and f be a character on G. In the situation I'm looking at f is either 1 or -1 at any point. Function f can be extended to ... 0answers 105 views ### Fourier transformation of a distribution We have no idea how to tackle the following Fourier transformation of a distribution:$$ \lim_{\epsilon\to0^+}\int_{-\infty}^{\infty}\mathrm{d} t\int_{\mathbb{R}^{d-1}} \mathrm{d}^{d-1}\vec{r} e^{-\... 0answers 224 views ### Singularity of L^1-solutions to elliptic PDEs on the puntured ball Let$\mathbb{B}$be the unit ball in$\mathbb{R}^n$. Then it is true that if$u\in L^1(\mathbb{B})$such that$\Delta(u)=0$on$\mathbb{B}\backslash\{0\}$, then$\Delta(u)$, as a distribution on$\...
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I know that the order of any distribution of compact support is finite. Is this true in the case of ultra distribution of compact support ( dual of Denjoy-Carleman space)?
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### Constrain representation of tempered distribution

This is a follow-up to this question. Let $T$ be a tempered distribution on $\mathbb{R}^d$. Then there is a multiindex $\alpha \in \mathbb{N}_0^d$, an $n \in \mathbb{N}_0$ and a bounded continuous ...
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### Prove that a given distribution is tempered

Suppose I have a distribution $E$ such that $\phi \ast E$ is square-integrable for all $\phi \in C_c^\infty \left( \mathbb{R}^d \right)$. Is it possible to prove that $E$ is tempered? It seems ...
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### Does this formula correspond to a series representation of the Dirac delta function $\delta(x)$?

Consider the following formula which defines a piece-wise function which I believe corresponds to a series representation for the Dirac delta function $\delta(x)$. The parameter $f$ is the evaluation ...
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### Distribution boundary value of analytic function and wave front sets

Assume $f(z)$ is analytic in the tube domain $\mathbb R^n\oplus iC$, where $C\subset \mathbb R^n$ is a convex cone. Under the assumption $|f(x+iy)|\leq 1/|y|^k$, we know by a Theorem of Martineau (see ...
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### Function of moderate growth: history, motivation, and uses

I recently came across functions of moderate growth via this post and I was wondering, what are some concrete uses or applications of this space? Where does it appear and why was it introduced ...
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### $GL_1(\mathcal{E}'(\mathbb{R}))$ open in $\mathcal{E}'(\mathbb{R})$?

Let $\mathcal{E}'(\mathbb{R})$ be algebra of all compactly supported distributions on $\mathbb{R}$, equipped with the strong dual topology $\beta(\mathcal{E}',\mathcal{E})$, and with the usual ...
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Suppose $f: \mathcal{S}(\mathbb{R}^{d})^{n+1} \to \mathbb{C}$ is a continuous function. To each $\varphi \in \mathcal{S}(\mathbb{R}^{d})$, we can define the map $f[\varphi]: \mathcal{S}(\mathbb{R}^{d})... 2answers 183 views ### A question about homogeneous distribution A distribution in$\mathscr{S}^{\prime}\left(\mathbb{R}^{n}\right)$is called homogeneous of degree$\gamma \in \mathbb{C}$if for all$\lambda>0$and for all$\varphi \in \mathscr{S}\left(\mathbb{...
First of all, I am not sure of the terminology here, I am interested in the function $$F(x)=x|x|^{-d},x\in \mathbb{R}^d\setminus \{0\}$$ in dimension $d\geq 2$. I read somewhere that this is called ...