# 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 a classification of 2D projective convolution kernels?

Is there any classification of all distributions on $\mathbb{R}^2$ such that they are equal to the convolution with themselves? i.e. given a distribution $\gamma$ under which conditions
$$ \gamma\star\...

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### Why $\int_{S^{n-1}} |\hat{f}(w)|^2d\sigma(w) < \infty$?

Let $f\in L^p(\Bbb R^n)$ and $S^{n-1}$ be the Unit sphere.
Why
$\int_{S^{n-1}} |\hat{f}(w)|^2d\sigma(w)<\infty$ when $1<p<2$.
$\hat{f}$ is the Fourier transform fora function f.

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### Intermediate spaces of test functions between $\mathcal{S}$ and $\mathcal{D}$?

On $\mathbb{R}^n$, let $\mathcal{S}(\mathbb{R}^n)$ be the Schwartz space and $\mathcal{D}(\mathbb{R}^n)$ be the space of smooth, compactly supported functions.
According to p.145 of the book by Reed &...

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### Extracting each field operator as Wightman fields from a set of time-ordered products satisfying Eckmann-Epstein axioms

The paper by Eckmann-Epstein proves that Schwinger functions at "coinciding points" uniquely defines "time-ordered products".
In physics, these "time-ordered products" ...

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### Functions with derivatives growing at rate $r>0$

Fix a non-empty closed subset $\Omega\subset\mathbb{R}$.
Let $f:\mathbb{R}\to\mathbb{R}$ be smooth and such that $\sup_{x\in \Omega}\,|\partial^k f(x)|\lesssim k^r$ for some $r\ge 0$ for all $k\in \...

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### Tempered distributions at non-coinciding points and density of Schwartz functions

In the previous question, I find that situation is much less favorable than expected…. So I add more details to focus on the specific case I have in mind.
Let us consider the Schwartz space $\mathcal{...

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### $C^\infty$-coring

We know that there the so called smooth algebras also known as $C^\infty$-rings. They can play an important role in modern treatment of differential geometry. Is there a coring analogue?

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### Action of Hopf algebra of identity supported distributions on a Lie group

The Hopf algebra
of identity supported distributions on a lie group is cocommutative. It is well known that it is a group object in the category of cocommutative coalgebras. Is there a canonical ...

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### Does this distribution exist?

Assume there is a distribution in two variables $\mathcal{W}\in\mathcal{S}'(\mathbb{R}^2)$ with Fourier transform $\hat{\mathcal{W}}(\alpha,\beta)\equiv \int_{-\infty}^\infty e^{i(\alpha x+\beta y)} \...

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### A question about the eigenfunction method and the notion of solution - distributional solution

I have a question about how a passage was made in the calculation of passage (2.5) in the calculation below. To introduce context, the author in the paper (full work) is trying to demonstrate that ...

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### Wick product of free fields and wave front sets in the sense of Lars Hörmander

Let $\phi$ be the neutral, massive and free scalar field in $\mathbb{R}^4$. That is, $\phi$ is a tempered distribution whose values are unbounded operators on the Bosonic Fock space.
Note that the ...

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### Existence and uniqueness of solutions to a distributional ordinary differential equation

Suppose that $v$ is a distribution on the real line. Then under what conditions can I solve the differential equation
$$
\dot{x}(t) = v(x(t))
$$
which I might interpret as an integral equation
$$
-\...

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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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### Approximating a sequence of tempered distributions "uniformly" by Schwartz functions

This question has been motivated by the post making sense of distributions on the diagonal.
Let $T$ be a tempered distribution on $\mathbb{R}^2$ and $\eta$ be a given mollifier on $\mathbb{R}$. For $f ...

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### For a tempered distribution $F$ on $\mathbb{R}^2$, what exactly does it mean by $\lvert F(x,y) \rvert \leq \lvert x-y \rvert^{-n}$?

Let $F$ be a tempered distribution on $\mathbb{R}^2$ and $n \in \mathbb{N}$ be a fixed natural number. I wonder what exactly it means by
$\lvert F(x,y) \rvert \leq \lvert x-y \rvert^{-n}$ where $x,y \...

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### Construction of random tempered distributions

Let $(\xi_\phi)_{\phi \in L^2(\mathbb{R}_+ \times \mathbb{R}^d,\lambda_d)}$ be a collection of centered Gaussian processes on a probability space $(\Omega,\mathcal{F},P)$ such that $$\forall \phi \in ...

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### Is the Schwartz space a tame Frechet space?

I ran into the following definition of tame Frechet spaces and Nash-Moser therem.
It says that the space of smooth functions on a compact manifold is tame Frechet.
However, I wonder if
The Schwartz ...

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### Topology of ${\mathcal D}(\Omega)$ (space of test functions)

I have seen two approaches to the topology of ${\mathcal D}(\Omega)$:
(i) Let $K$ be a compact subset of $\Omega$ and consider the subset ${\mathcal D}_K(\Omega)$ of test functions with support ...

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### Is it possible to bound the L2 norm of the gradient of a divergent by the L2 norm of the Lapacian?

Is it possible to show for $u:\Omega\subset\mathbb{R}^3\rightarrow \mathbb{R}^3$
that
$$\||\nabla(\nabla\cdot u)|\|_2^2\leq C\||\Delta u|\|_2^2?$$
Here $\||f|\|_2$ is the norm in $(L^2(\Omega))^3$ and ...

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### How are distributions and divergent series summations related?

When we do Fourier analysis, we don't always get convergent series. A classic example comes from considering the Sawtooth function. It has Fourier Coefficients
$$s(x) = \frac{1}{2} + \sum_{n \neq 0} \...

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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
\begin{equation}
V \subset H \subset V'
\end{equation}
where $V'$ is the dual of $V$ and the inclusions are ...

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### Special Schwartz function on the positive interval

Is there a Schwartz function $\zeta(t)$, defined on $\mathbb{R}$, satisfying the following:
$\int \zeta(t)\: dt=1$,
$\int t^k \zeta(t)\: dt=0$ for all $k\geq 1$,
$\operatorname{supp}(\zeta)\subset (0,...

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### Examples of Borel probability measures on the Schwartz function space?

Let $\mathcal{S}(\mathbb{R}^d)$ be the Frechet space of Schwartz functions on $\mathbb{R}^n$. Its dual space $\mathcal{S}'(\mathbb{R}^d)$ is the space of tempered distributions.
Minlos Theorem as ...

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### Making sense of the limit $\lim\limits_{x \to y} T(x,y) $ for a tempered distribution $T$ on $\mathbb{R}^{2n}$

I already posted a similar question on MO and looked into the references therein.
However, I cannot find a satisfactory answer for my question..So I ask here again in a more refined form.
Let $T \in \...

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### Is the test function topology a Mackey topology?

I am a physicist, and I have lately been thinking about distributions as they appear in quantum field theory. In the standard development of the theory of distributions, one considers the space $C^{\...

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### Derive distributional inequalities from pointwise estimates

My question is how to prove the following claim:
Suppose that $E$ is an algebraic set in $\mathbb{R}^n (n\ge3)$ with dimension $\le n-2$, and $u$ is locally Lipschitz continuous
on $\mathbb{R}^n$. If ...

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### Limit of a distribution using Hörmander’s theorem

Let $\alpha \in \mathbb{C}$. I want to prove that
$$ (e^{i2\theta}\xi_1^2 + \xi_2^2 + \dots + \xi_n^2)^{-\alpha} \longrightarrow (Q(\xi)-i0)^{-\alpha}, $$
in $D’(\mathbb{R}^n\setminus \left\{0\right\})...

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### Is this a well known space? Perhaps homogeneous Sobolev-like space?

The homogeneous Sobolev space $\dot H^s(\mathbb{R}^n) $ is often defined as the closure of $\mathcal{S}(\mathbb{R}^n)$ under the norm
$$ || |\omega|^s \widehat{f} ||_{L^2(\mathbb{R}^d)} =\int_{\...

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### Non-Schwartz test functions for the explicit formula for L-functions

The statements of the explicit formula for L-functions that I am aware of require the test function to be a Schwartz function (see, e.g., equation (4.11) in Section 4 of Low lying zeros of families of ...

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### Is there an infinite dimensional Stein's lemma?

Classical Stein's lemma says that if $\mathbf{X}$ is a centered Gaussian random vector and $g$ is a function which is nice enough, we have
$$
\mathbb{E} \, X_i \, g ( \mathbf{X} )
= \sum_k \...

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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
\begin{equation}
\lVert F(f) \rVert \leq \lVert f \rVert
\end{equation}
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 ...

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### 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"...