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.
361
questions
3
votes
1
answer
196
views
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 ...
3
votes
1
answer
446
views
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 $\langle A, a\rangle$ ...
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 ...
1
vote
0
answers
103
views
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_{...
25
votes
5
answers
3k
views
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 ...
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 (...
2
votes
2
answers
383
views
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 ...
2
votes
0
answers
94
views
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 ...
3
votes
1
answer
239
views
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:
$$\lim_{t\to 0} \int_{-\infty}^\infty \left(e^{-\log(4\pi i t)/2} e^{ik^2/4t} -\...
2
votes
2
answers
426
views
Approximating a subclass of $L^2(\mathbb{R})$ by Schwartz functions within similar subclass
It is well-known that real valued Schwartz functions on $\mathbb{R}_+$ $\mathcal{S}(\mathbb{R}_+)$ are dense in the set of square integrable functions on $\mathbb{R}_+$ $L^2(\mathbb{R}_+)$. We can ...
3
votes
1
answer
147
views
Is every compactly embedded Banach subspace of the space of distributions contained in some negative Holder space?
Let $F(=(C_c^\infty)^\ast,S^\ast)$ be the space of (tempered) distributions. Let $B\hookrightarrow F$ be a compactly embedded Banach subspace. Is it true that $B\subset C^{-\alpha}$ for some $\alpha&...
6
votes
0
answers
139
views
Does $f \in L^1([0,T]; S'(\mathbb R^n))$ define a $(1+n)$-dimensional distribution?
Let $f : [0,T] \rightarrow S'(\mathbb R^n)$ be a family of tempered distributions satisfying
$$\langle f(t), \phi \rangle \in L^1([0,T])$$
for any Schwartz function $\phi \in S(\mathbb R^n)$.
Does $f$ ...
7
votes
0
answers
133
views
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}(\...
4
votes
0
answers
136
views
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}$ ...
2
votes
0
answers
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 \...
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 ...
6
votes
1
answer
394
views
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 ...
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 ...
1
vote
0
answers
193
views
"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 ...
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 ...
1
vote
0
answers
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)}}$$
Formally $g' = -\frac{1}{f}g$.
How can I justify this ...
1
vote
1
answer
309
views
How to understand subharmonic functions, distributions, and measure?
Sorry if this turns out to be a silly question, but I am having difficulties in both understanding it and finding other references for it. I hope that someone can clear my concepts here on overflow.
...
5
votes
1
answer
445
views
English translation of Schwartz's papers on vector-valued distributions
I am interested in systematically studying the theory of vector-valued distributions. The original two papers due to Laurent Schwartz entitled Théorie des distributions à valeurs vectorielles. I & ...
2
votes
0
answers
50
views
Is this Beppo-Levi curl space a Banach space?
Let us define the quotient space:
$$ 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)....
-4
votes
1
answer
357
views
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
$$\delta(z) = \...
4
votes
1
answer
235
views
Literature on the product of two distributions satisfying the Hörmander condition
I am currently studying some basic questions concerning the product $uv\in \mathscr{D}'(\mathbb R^n)$ of two Schwartz distributions $u,v\in \mathscr{D}'(\mathbb R^n)$ satisfying the Hörmander ...
4
votes
0
answers
104
views
Poincaré's Lemma in the space of tempered distributions
It is well known that if $f\in \mathcal{D}'(\mathbb{R}^3,\mathbb{R}^3)$ and $\textbf{curl} f= 0$ then there exists a $u\in \mathcal{D}'(\mathbb{R}^3)$ such that $\nabla u = f$.
Question. Does the ...
2
votes
0
answers
85
views
Fourier Transform ; half space elliptic baby problem
I am attempting to look at some Liouville type theorems via a Fourier analysis approach and
after looking at a baby problem I seem to be very confused. I assume this doesn't count as a research ...
4
votes
1
answer
308
views
Approximating compactly supported $L^2$ functions with Schwartz functions "from within"?
It is well known that the class of Schwartz functions $\mathcal{S}$ in dense in all $L^p$ spaces therefore for each $f \in L^2$ there exists a sequence of Schwartz functions $(f_k)$ such that $\lVert ...
5
votes
0
answers
285
views
Feynman path integral and Wilsonian renormalization
Everything below is to be viewed in the Euclidean setting with $d$ dimensions and all measures are understood to be Borel measures.
The usual problem of Quantum Field Theory is to make sense of ...
2
votes
0
answers
129
views
Support of a fundamental solution of wave equation
The solution of the wave equation
$$
\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^{...
0
votes
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 ...
6
votes
0
answers
154
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^{-\...
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 $\...
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)?
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 $...
2
votes
1
answer
290
views
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 ...
5
votes
0
answers
99
views
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 ...
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. ...
14
votes
0
answers
672
views
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 "...
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 ...
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 ...
14
votes
1
answer
1k
views
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, ...
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 ...
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
...
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 $E$. ...
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 $...
1
vote
0
answers
53
views
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 ...
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 ...
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 ...