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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Smooth cut-off in homogeneous Besov space

Given a Littlewood-Paley decomposition $$1 = \chi(\xi) + \sum_{j \geq 0}\varphi(2^{-j} \xi), \quad \xi \in \mathbb R^n$$ where $\chi$ is smooth, supported on a ball, and $\varphi$ is smooth, supported ...
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How to rigorously differentiate the convolution of a distribution and a $L^2$ function?

I want to prove the following: (Here, $W^{2,2}$ is a Sobolev space as defined in Evans, chapter 5; $S$ is a Schwartz space; and if $A$ is a distribution and $a$ a function, then $\langle A, a\rangle$ ...
Maximilian Janisch's user avatar
1 vote
1 answer
323 views

Spectral theorem and diagonal expansion for self adjoint operators

Asked by a physicist: In physics we use the fact that, given a self adjoint operator $A$, every element in Hilbert space can be written as "generalized linear combination" of the eigenstates ...
Rosario's user avatar
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1 vote
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Does convergence of tempered distributions implies convergence in $\mathcal{S}(\mathbb{R}^4,\mathbb{R})/\mathcal{S}_{0}$?

We can define the following symmetric semi-definite positive bi-linear form on $\mathcal{S}(\mathbb{R}^{4},\mathbb{R})$ with values in $\mathbb{C}$, \begin{equation}\label{prodintespaciales} (h_{...
Gabriel Palau's user avatar
25 votes
5 answers
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Why are distributions "tempered"?

Google N-Gram shows that both "tempered distribution" and "temperate distribution" are used in English, but the first version significantly prevails, and usage of the second term ...
Alexandre Eremenko's user avatar
5 votes
3 answers
2k views

Fourier transform of periodic distributions

Following M. Ruzhansky and V. Turunen's book Pseudo-Differential Operators and Symmetries, in Chapter 3, Definition 3.1.25 (page 304), the space of periodic distributions is defined as follows (...
spaceman's user avatar
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2 answers
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Does this operator have a continuous, localized eigenfunction with negative eigenvalue?

I am looking at a class of operators $$ L[f](x)=af_{xxxx}-bf_{xx}+\frac{d}{dx}(\delta(x)f_x) $$ , a<0,b<0, on the real line, where $\delta$ is Dirac-delta. I am interested in ruling out the ...
mathamphetamine's user avatar
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Evolution PDE in dual space : Generalization of a result of Gelfand

The following result is proved in "Generalized functions", Volume 3, Chapter 2.2 by Gelfand : Let $\Phi$ be a Fréchet space with dual space $\Phi'$ endowed with the weak topology. For ...
Desura's user avatar
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How to prove that this one-parameter family of distributions converges to the Dirac measure?

While trying to understand a proof in a paper, I came upon the following a calculation needing the following identity: $$\lim_{t\to 0} \int_{-\infty}^\infty \left(e^{-\log(4\pi i t)/2} e^{ik^2/4t} -\...
Dispersion's user avatar
2 votes
2 answers
426 views

Approximating a subclass of $L^2(\mathbb{R})$ by Schwartz functions within similar subclass

It is well-known that real valued Schwartz functions on $\mathbb{R}_+$ $\mathcal{S}(\mathbb{R}_+)$ are dense in the set of square integrable functions on $\mathbb{R}_+$ $L^2(\mathbb{R}_+)$. We can ...
Plussoyeur's user avatar
3 votes
1 answer
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Is every compactly embedded Banach subspace of the space of distributions contained in some negative Holder space?

Let $F(=(C_c^\infty)^\ast,S^\ast)$ be the space of (tempered) distributions. Let $B\hookrightarrow F$ be a compactly embedded Banach subspace. Is it true that $B\subset C^{-\alpha}$ for some $\alpha&...
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6 votes
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139 views

Does $f \in L^1([0,T]; S'(\mathbb R^n))$ define a $(1+n)$-dimensional distribution?

Let $f : [0,T] \rightarrow S'(\mathbb R^n)$ be a family of tempered distributions satisfying $$\langle f(t), \phi \rangle \in L^1([0,T])$$ for any Schwartz function $\phi \in S(\mathbb R^n)$. Does $f$ ...
Desura's user avatar
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Characterization of tempered distributions from tempered sequences

Let $\mathcal{D}(\mathbb{R})$ be the space of compactly supported and infinitely smooth functions for its usual topology. Let $\mathcal{D}'(\mathbb{R})$ be the topological dual of $\mathcal{D}(\...
Goulifet's user avatar
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If theorem valid for compactly supported distribution then is it also valid for tempered distribution?

I have seen many theorem which Author wanted to prove for tempered distribution, but without saying anything proves for compactly supported distribution. For instance, Theorem: Any $A \in \Psi^{m}$ ...
Curious student's user avatar
2 votes
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178 views

Convergence in $S'(\mathbb R^d)$ of the paraproduct $\dot{T}_uv$

Let $B = B(0,4/3)$, $C = \{x \in \mathbb R^d : 3/4 \leq \|x\|_2 \leq 8/3\}$ and $\tilde{C} = \{x \in \mathbb R^d : 1/12 \leq \|x\|_2 \leq 10/3\}$. For a fixed Littlewood-Paley decomposition $\chi \in \...
Desura's user avatar
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6 votes
2 answers
328 views

A smooth function such that the second derivative of its absolute value is a distribution of positive order

Let $f\in C^\infty(\mathbb R;\mathbb R)$ and let us define $g(x)=\vert f(x)\vert$. It is easy to verify that $g$ is locally Lipschitz-continuous function, but I would like to find an example of a ...
Bazin's user avatar
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6 votes
1 answer
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Is the tensor product of distributions a continuous bilinear map with respect to the weak topology?

Let $X$ and $Y$ be smooth manifolds. The map $\mathcal{D}'(X)\times\mathcal{D}'(Y)\to\mathcal{D}'(X\times Y)$ given by $(S,T)\mapsto S\boxtimes T$ is continuous with respect to the strong topology. Is ...
user449595's user avatar
16 votes
2 answers
1k views

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 ...
YuerWu's user avatar
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"Potential" for a divergence-free distribution

Edit: I would like to reopen this question since the linked potential duplicate question is not useful in showing that we get a tempered distribution as the potential, and I have found no easy way to ...
Maximilian Janisch's user avatar
4 votes
1 answer
434 views

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 ...
Maximilian Janisch's user avatar
1 vote
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88 views

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 ...
user895874's user avatar
1 vote
1 answer
309 views

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. ...
ldgo's user avatar
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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 & ...
genfuntranslate's user avatar
2 votes
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50 views

Is this Beppo-Levi curl space a Banach space?

Let us define the quotient space: $$ V = \{ \mathbf{u} \in L^2_{loc}(\mathbb{R}^3; \mathbb{R}^3) : \operatorname{curl} \mathbf u \in L^2(\mathbb{R}^3; \mathbb{R}^3) \} / \nabla H^1_{loc}(\mathbb{R}^3)....
GaC's user avatar
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1 answer
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Is delta function symmetric against real axis? [closed]

Is $\delta\left(a+bi\right)=\delta\left(a-bi\right)$? I wonder whether Dirac Delta (as defined via Fourier transform) is symmetric against the real axis. We can write Delta function as $$\delta(z) = \...
Anixx's user avatar
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4 votes
1 answer
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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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104 views

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 ...
Kosh's user avatar
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0 answers
85 views

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 ...
Math604's user avatar
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4 votes
1 answer
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Approximating compactly supported $L^2$ functions with Schwartz functions "from within"?

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 ...
Dominic Shillingford's user avatar
5 votes
0 answers
285 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 ...
iolo's user avatar
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2 votes
0 answers
129 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^{...
evedel's user avatar
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0 answers
51 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 ...
alesia's user avatar
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6 votes
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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^{-\...
Y.Okuyama's user avatar
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2 votes
0 answers
235 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 $\...
T. Le's user avatar
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1 vote
0 answers
65 views

Order of ultradistribution

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)?
Jem Y's user avatar
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2 votes
0 answers
125 views

Mixed partial derivatives of planar functions converging to delta distribution

Given a sequence $(f_k)_{k\in\mathbb{N}}\subset C^2(\mathbb{R}^2)$ of strictly positive functions $f_k\equiv f_k(x,y)$ with $\|f_k(x,\cdot)\|_{L^1}=1$ for all $x\in\mathbb{R}$ and such that for each $...
cts12's user avatar
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2 votes
1 answer
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About Dirac function

In Vladimirov's book "A Collection of Problems on the Equations of Mathematical Physics", p129, 11.16, there is a equality about Dirac function, which is the fundamental solution of three ...
WPJ's user avatar
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0 answers
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Decomposition of the Schwartz space as a representation for the orthogonal group

The Schwartz space $\mathcal{S}(\mathbb{R}^n)$ is naturally a $O_n(\mathbb{R})$-representation. I'm assuming that this is a relatively well-behaved representation among the infinite-dimensional ones ...
Johannes Hahn's user avatar
5 votes
0 answers
281 views

Wightman reconstruction theorem-details of the proof

First of all forgive me if this question is not well suited for this forum: it is motivated by physics however after all my concerns are mathematical so I hope it would be appropriate to post it here. ...
truebaran's user avatar
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14 votes
0 answers
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strong topologies on $C_c^\infty$

UPDATE (27/08/2020): I realized after a comment from Jochen Wengenroth that there was at least one false premise behind my question, owing to the fact that analysts sometimes use the words "...
Chris Wendl's user avatar
4 votes
1 answer
174 views

Are nuclear spaces used in creating variant theories of distributions?

Laurent Schwartz proved his Kernel Theorem in 1952  to justify extending his theory of distributions to several variables. Then he and Jean Dieudonne gave Alexander Grothendieck the assignment to ...
Colin McLarty's user avatar
5 votes
0 answers
264 views

Riesz Representation Theorem for $L^2(\mathbb{R}) \oplus L^2(\mathbb{T})$?

The spaces $L^2(\mathbb{R})$ (square-integrable functions) and $L^2(\mathbb{T})$ (1-periodic square-integrable functions, considered over the real line $\mathbb{R}$) are two subspaces of the space of ...
Goulifet's user avatar
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14 votes
1 answer
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What is the analytic continuation of $\varphi(s)=\sum_{n \ge 1} e^{-n^s}?$

My research has lead me to the following function that I'm trying to continue. 3 Months ago I posted this question to MSE, and have placed 3 bounties on the question, but haven't received an answer, ...
geocalc33's user avatar
5 votes
1 answer
162 views

Critical Smoothness on Besov Spaces $B^s_{p}$: how does it evolved with $p$?

We denote by $B_{p}^s(\mathbb{T}) := B_{p,p}^s(\mathbb{T})$ the Besov space over the circle $\mathbb{T}$ with parameters $p=q \in (0, \infty]$ and smoothness $s \in \mathbb{R}$. For $p>0$ fixed and ...
Goulifet's user avatar
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11 votes
2 answers
829 views

How do you know that you have succeeded-Constructive Quantum Field Theory and Lagrangian

Quantum Field Theory is a branch of mathematical physics which is begging for a better understanding. In fact there are no rigorous constructions of interacting QFT in four dimensions. By a rigorous ...
truebaran's user avatar
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5 votes
1 answer
330 views

When is a distribution having a finite support actually zero?

Let $D$ be a differential operator with smooth coefficients in $\mathbb{R}^n$. Suppose $E$ is a bounded open set in $\mathbb{R}^n$. Suppose $u$ is a function that is smooth up to the boundary of $E$. ...
T. Le's user avatar
  • 562
3 votes
1 answer
116 views

Existence of a special function

Consider a $C^2$ bounded domain $D$ of $\mathbb{R}^d$. Let $b \subset \partial D$ a non-empty part of the boundary. Let $n(x)$ be the unit outward vector on $\partial D$. Is there any smooth function $...
MathGeo's user avatar
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1 vote
0 answers
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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 ...
iolo's user avatar
  • 611
7 votes
2 answers
958 views

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 ...
iolo's user avatar
  • 611
5 votes
3 answers
1k views

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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