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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 ...
zoran  Vicovic's user avatar
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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 ...
Qfwfq's user avatar
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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\...
Nicolas Medina Sanchez's user avatar
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1 answer
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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.
Edward's user avatar
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7 votes
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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 &...
Isaac's user avatar
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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" ...
Isaac's user avatar
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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 \...
Math_Newbie's user avatar
2 votes
1 answer
144 views

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 $\mathcal{...
Isaac's user avatar
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$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 ...
Lefevres's user avatar
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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)} \...
Nicolas Medina Sanchez's user avatar
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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 ...
Ilovemath's user avatar
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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 ...
Isaac's user avatar
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3 answers
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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 $$ -\...
cheshircat's user avatar
2 votes
2 answers
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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, ...
cheshircat's user avatar
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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 ...
Isaac's user avatar
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2 answers
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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 \...
Isaac's user avatar
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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 ...
mathex's user avatar
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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 ...
Isaac's user avatar
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2 votes
1 answer
158 views

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 ...
olih's user avatar
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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 ...
Alberto Leandro's user avatar
3 votes
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114 views

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} \...
Caleb Briggs's user avatar
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1 answer
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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{...
Filippo's user avatar
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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 ...
Isaac's user avatar
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8 votes
1 answer
349 views

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,...
SnowRabbit's user avatar
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274 views

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 ...
Isaac's user avatar
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2 votes
2 answers
260 views

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 \...
Isaac's user avatar
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4 votes
0 answers
169 views

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^{\...
Jon's user avatar
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1 answer
142 views

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 ...
William's user avatar
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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\})...
zarathustra's user avatar
1 vote
0 answers
185 views

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_{\...
Dan1618's user avatar
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3 votes
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204 views

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 ...
Tristan Phillips's user avatar
8 votes
1 answer
396 views

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 \...
tsnao's user avatar
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1 vote
1 answer
112 views

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 ...
Isaac's user avatar
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1 vote
0 answers
60 views

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-...
Isaac's user avatar
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2 votes
0 answers
40 views

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^{\...
Ali's user avatar
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3 votes
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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 ...
Isaac's user avatar
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6 votes
1 answer
223 views

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 ...
G. Blaickner's user avatar
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3 votes
4 answers
329 views

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 ...
Colin McLarty's user avatar
2 votes
1 answer
58 views

$K *g_n$ 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 ...
Isaac's user avatar
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2 votes
2 answers
192 views

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 $f^{(m)}_n$ ...
Isaac's user avatar
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2 votes
0 answers
100 views

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 ...
B.Hueber's user avatar
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2 votes
0 answers
96 views

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://...
fast_and_fourier's user avatar
1 vote
1 answer
138 views

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 ...
user72829's user avatar
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3 votes
0 answers
80 views

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-...
Isaac's user avatar
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1 vote
1 answer
112 views

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 ...
Isaac's user avatar
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2 votes
0 answers
56 views

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 ...
Isaac's user avatar
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4 votes
1 answer
124 views

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 ...
백주상's user avatar
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4 votes
1 answer
331 views

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 ...
Isaac's user avatar
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0 votes
0 answers
80 views

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"...
Isaac's user avatar
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