# 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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### The limit is in $S(\Bbb R^n)$ [closed]

Let $f_n(x)\in S(\Bbb R^n)$ such that $\sum_{n\in\Bbb N} f_n(x)$ converge in $C^\infty(\Bbb R^n)$. Let $S(x)=\sum^\infty_{n=0} f_n(x)$. My question is $S\in S(\Bbb R^n)$. Thank you in advance. Proof ...
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### Is $1/F$ Schwartz if $F$ is "reverse Schwartz"?

Let's call a positive function $F:\mathbb{R}\to\mathbb{R}$ "reverse Schwartz" if $F$ is smooth and $$\forall n \forall k,\quad\lim_{x\to\infty}\frac{|x|^n}{|\partial_x^k F(x)|}=0\quad .$$ In ...
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### Is there any work on distributional vector fields?

I know that people think about weak solutions to PDEs by turning a differential equation into an integral equation. Have people studied any analogue to this where we start with an integral equation, ...
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### Proof that generalized Laplacian is essentially self-adjoint (Heat Kernels and Dirac Operators)

According to Proposition 2.33 in Heat Kernels and Dirac Operators each symmetric generalized Laplacian $H$ is essentially self-adjoint. This is an immediate consequence of the fact that \begin{...
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### Finding an element of Gelfand triple with a designated time derivative

Let $V$ be a real separable Banach space and $H$ be a real separable Hilbert space such that $$V \subset H \subset V'$$ where $V'$ is the dual of $V$ and the inclusions are ...
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### A Gaussian measure $\mu$ on $\mathcal{E}'(S^1)$ by Minlos theorem and its value for Sobolev spaces $H^{\alpha}(S^1)$

I posted this question on ME as "A Gaussian measure on $\mathcal{E}'(S^1)$ by Minlos Theorem and its value for $L^2(S^1)$", but it seems much more nontrivial than I expected... so, I post an ...
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### Infinite dimensional version of the Laplace transform and Gaussian integrals

This question is somehow related to my previous one Convergence of the Gaussian integral on $\mathcal{E}'$ for a mapping supported on $L^2$ Let $F : L^2(S^1) \to L^2(S^1)$ be a (nonlinear) Borel-...
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### Blow up for certain classes of distributions

Let $\mathbb D$ be the open unit disc centered at the origin and let $u \in H^{-N}(\mathbb D)$ be a distribution for some natural number $N>0$. Suppose that u|_{\mathbb D\setminus \{0\}} \in C^{\...
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### Convergence of the Gaussian integral on $\mathcal{E}'$ for a mapping supported on $L^2$

Let $F : L^2(S^1) \to L^2(S^1)$ be a (nonlinear) mapping such that $$\lVert F(f) \rVert \leq \lVert f \rVert$$ for all $f \in L^2(S^1)$. For the space of smooth periodic ...
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### Existence of adjoint operators on manifolds

Let $(M,g)$ be an oriented Riemannian manifold and $V$ a finite-rank vector bundle equipped with a non-degenerate bundle metric $\langle\cdot,\cdot\rangle_{V}$. This bundle metric, in turn, gives rise ...
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### What real distributions solve $f'=0$? [closed]

I mean specifically real-valued Schwartz distributions on the real line.  That is linear functionals  on $C^{\infty}_c(\mathbb{R})$ continuous in the canonical LF topology.  My question is, what are ...
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### $K *g_n$ converges in the topology of smooth functions, $K$ approximates $\delta(x)$ and $g_n$ is a.e convergent to $g$, then regularity of $g$?

This question is continuation from If $K *g_n$ converges in the Fréchet topology of smooth functions and $K$ approximates $\delta(x)$, is $g_n$ itself convergent? - revised. As before, let us ...
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### If $K *g_n$ converges in the Fréchet topology of smooth functions and $K$ approximates $\delta(x)$, is $g_n$ itself convergent? - revised

Let us consider the Fréchet space $C^\infty\Bigl([0,1],\mathbb{R} \Bigr)$ of real-valued, periodic smooth functions. That is, $f_n \to f$ in $C^\infty\Bigl([0,1],\mathbb{R} \Bigr)$ if $f^{(m)}_n$ ...
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### Schwartz kernel theorem for restricted operators

Let $(M,g)$ be a smooth Riemannian manifold. The celabrated kernel theorem of Schwartz shows that for any linear and continuous operator $A:C_{c}^{\infty}(M)\to C^{\infty}(M)$, there exists a ...
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### Why do we work on homogeneous Besov/Triebel-Lizorkin spaces?

This question is mainly for understanding the history behind homogeneous spaces. There is extensive literature on Besov and Triebel-Lizorkin spaces. For instance, see the standard textbook: https://...
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### Banach space valued distributions and test functions

Let $A,B,C$ be Banach spaces and $m\,:\,A\times B\to C$ be a bilinear map such that $\|m(a,b)\|\leq \textrm{const}\,\|a\|\|b\|$. We denote by $\mathcal{S}(\mathbb{R}^d)$ be the standard space of ...
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### Does the Minlos theorem work for real-valued cases as well?

Let $\mathcal{E}(\mathbb{T}^3, \mathbb{R})$ be the Frechet space of real-valued smooth periodic functions on $\mathbb{R}^3$. Here, $\mathbb{T}^3$ is the $3$-dimensional torus. Let us define a real-...
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### The notion of "Admissible" and "Permitted" in the context of convolution with distributions and hypocontinuity

I am reading the paper "On Convolutions" (1958) and have encountered the notion of "Admissible" and "Permitted" spaces. In p.17-18 of the above paper, it says that an ...
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### Regularity on $\mathbb{T}^3$ of the "functional average" of a map $S : C^\infty(\mathbb{T}^3, \mathbb{R}) \to L^2(\mathbb{T}^3, \mathbb{R})$

For simplicity, let $C^\infty(\mathbb{T}^3, \mathbb{R})$ be the real Frechet space of periodic smooth functions on $\mathbb{R}^3$. Here, $\mathbb{T}^3$ is the $3$-dimensional torus. For a fixed ...
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### Question about calculation in Schwartz space

While reading a paper Hengang Li and Weiping Yan - Asymptotic stability and instability of explicit self-similar waves for a class of nonlinear shallow water equations, I experienced that my ...
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### How do we give a rigorous mathematical meaning to expressions like $\delta^4(0)$ or $\lim\limits_{x \to y} \delta^4(x-y)$?
The question is as in the title. In QFT literature, $\delta^4(0)$ is said to stand for the volume of entire $\mathbb{R}^4$, where $\delta^4(x)$ is the $4-$dimensional delta function. Or when defining ...
### Is it possible to continuously embed $C^\infty(\mathbb{T}^n)$ as a vector space into $\mathcal{D}(\mathbb{R}^n)$ by some "inverse" of periodization?
Let $\mathbb{T}^n$ be the $n-$dimensional torus and $C^\infty(\mathbb{T}^n)$ be the Frechet space of smooth periodic functions on $\mathbb{R}^n$. According to p.298 of Folland "Real Analysis"...