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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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 "...
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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}(\...
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Squaring random Schwartz distributions
Let $\mu$ denote the centered Gaussian measure on $S'(\mathbb{R}^d)$ with covariance
$$
\mathbb{E}
[\phi(f)\phi(g)]=\int_{\mathbb{R}^d} \frac{\overline{\widehat{f}(\xi)}
\widehat{g}(\xi)}{|\xi|^{d-2[\...
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Product of distributions under wavefront set condition is zero
Assume $u, v \in \mathcal{D}'(\mathbb{R}^n)$ are distributions with compact support. Denote by $\operatorname{WF}(\bullet) \subset T^*\mathbb{R}^n \setminus 0$ the wavefront set of a distribution $\...
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Does $f \in L^1([0,T]; S'(\mathbb R^n))$ define a $(1+n)$-dimensional distribution?
Let $f : [0,T] \rightarrow S'(\mathbb R^n)$ be a family of tempered distributions satisfying
$$\langle f(t), \phi \rangle \in L^1([0,T])$$
for any Schwartz function $\phi \in S(\mathbb R^n)$.
Does $f$ ...
6
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154
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Fourier transformation of a distribution
We have no idea how to tackle the following Fourier transformation of a distribution:
$$
\lim_{\epsilon\to0^+}\int_{-\infty}^{\infty}\mathrm{d} t\int_{\mathbb{R}^{d-1}} \mathrm{d}^{d-1}\vec{r} e^{-\...
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Uniform estimates of Fourier transform of tempered functions with parameters
Consider the following function in $\mathbb{R}^3$:
$$
f_t(x)=(1+|x|^2)^{-\alpha}e^{-g(x)t},\,\,\,\,\, \text{where}\,\, g(x)=\frac{x^2_1\cdot x^2_2}{1+|x|^2},
$$
where $\frac{1}{2}<\alpha<1$, and ...
6
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Norms and distributions
Question 1. Is there a nice or explicit way to describe the class of all distributions (generalized functions) $\mu$ on the $n$-sphere $S^n \subset \mathbb{R}^{n+1}$ for which the function
$$
F(v) := \...
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distributions on Lie groups and representations
Let $G$ be a Lie group and $\pi$ a continuous action on $V$, a Fréchet space.
This action induces a representation of the space of compactly supported functions, $C_c(G)$, with convolution as product ...
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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 ...
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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 ...
5
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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. ...
5
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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 ...
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Is polar decomposition of a smooth map Sobolev?
Motivation:
Let $\mathbb{D}^2$ be the closed unit disk. I am studying the "elastic energy" functional $E(f)=\int_{\mathbb{D}^2} \text{dist}^2(df,\text{SO}_2)$, where $f \in C^{\infty}(\mathbb{D}^2,\...
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Geometric characterization of Silva distributions
There is a well known geometric characterization of tempered distributions on $\mathbb{R}^n$.
A distribution $T\in \mathcal{D}'(\mathbb{R}^n)$ is an element of $\mathcal{S}'(\mathbb{R}^n)$ if and ...
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Harish-Chandra's submersive principle on closed subsets
Harish-Chandra's submersion principle says the following. Let $X,Y$ be two manifolds of dimensions $m$ and $n$ respectively. Let $\pi: X\rightarrow Y$ be a surjective smooth map which is submersive ...
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Differential operators acting on the Schwartz space
I asked a similar question on math stack exchange but didn't get an answer so I will try to ask it here. Any help/suggestion is most than welcome!
Let $D$ be a linear differential operator with ...
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Support of a Measure with Characteristic Functional Continuous in $L_p$, $1\leq p <2$?
Let $\mathcal{S}(\mathbb{R})$ be the space of smooth and rapidly decaying functions and $\mathcal{S}'(\mathbb{R})$ its dual, the space of tempered distributions. Let $\mathscr{P}$ be a probability ...
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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^{\...
4
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203
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Conormal distributions and the wave front set
Let $X$ be a smooth closed manifold and $Y$ a regular submanifold. For all conormal distributions at $Y$ on $X$, their wave front set is contained in the conormal bundle of $Y$. Is the reciprocal true?...
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Holonomic distributions in the analytic setting
We are interested in a reference to the notion of Holonomic distributions in the real analytic setting. Namely, distributions that generate a Holonomic D-module under the action of the algebra of ...
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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}$ ...
4
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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 ...
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462
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Applications and main properties of hyperfunctions
I am trying to get familiar with hyperfunctions, and I do have some familiarity with the classical theory of distributions.
I am wondering whether hyperfunctions have any advantages over ...
4
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Define the space of distributions with algebraic decay?
A tempered distribution $u\in \mathcal{S}'(\mathbb{R})$ is said to be rapidly decreasing if for every $f \in \mathcal{S}(\mathbb{R})$, $u*f \in \mathcal{S}(\mathbb{R})$.
One rough way to motivate ...
4
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Convergence of Schwartz Kernels
I read this question, and I would like to ask the opposite: Assume that I have a sequence of smoothing operators $(P_n)$ with (hence smooth) kernels $(p_n)$ converging strongly to some smoothing ...
4
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Is there an analogue of distributions in characteristic p?
Some motivation: When working over $\mathbb{C}$, distributions (in the sense of generalized functions) act as natural generators for $D$-modules (in the sense that any regular holonomic $D$-module is ...
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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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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 ...
3
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answers
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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-...
3
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Conditions ensuring that the paraproduct remainder is well-defined
In short, my question is: are there conditions that one can impose on two tempered distributions $u$ and $v$ that will guarantee that the paraproduct remainder $R(u,v)$ is well-defined and is "...
3
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Definition clarification: "regular directed distributions"
(I asked this question on math.stackexchange (see here) but didn't receive any reaction, hence I try it here. If it does not fit within here, just let me know in the comments.)
In the definition of ...
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Have there been recent developments of Booker's approach to L-functions as distributions?
Andrew Booker introduced a framework to study L-functions through distributions in https://arxiv.org/abs/1308.3067v2. This allowed him and others to get new results about zeros of automorphic L-...
3
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Regularity of fiber integration between complex analytic spaces
Let $f:X\rightarrow Y$ be a flat surjective morphism between reduced complex analytic spaces. Assume that $Y$ is locally irreducible (i.e. unibranch).
We assume that $X$ (resp. $Y$) is pure-...
3
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404
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de Rham currents/distributions on manifolds with boundaries
My main source for currents and distribution theory on manifolds in general is de Rham's Differentiable Manifolds. To recap, let $M$ be a smooth, $m$ dimensional real manifold without boundary. De ...
3
votes
1
answer
196
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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 ...
3
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answers
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Entire analytic functions with entire analytic Fourier transform, and corresponding distributions
I'm interested in the Fourier transform on a space of distributions that includes more than the usual tempered distributions, and in particular allows for $\delta$-distributions supported at complex ...
3
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174
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Question on de Rham complex with distributional coefficients
Let $X$ be a smooth manifold (usually assumed to be paracompact). Let us denote by $\underline{\Omega}^{p,-\infty}_X$ the sheaf of real valued $p$-forms with distributional coefficients in the ...
3
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(Non-)Existence of certain invariant distributions on a p-adic space
Following Bernstein-Zelevinski, an $\ell$-space is a Hausdorff, locally compact totally disconnected topological space. For an $\ell$-space $X$, denote $S(X)$ the space of Bruhat-Schwartz functions on ...
3
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Extension of Paley-Wiener-Schwartz theorem to vector-valued distributions
Let $H_{j} := (H_{j}, \| \cdot \|_{H_{j}} ), j=0,1$ be a Hilbert space, and set
\begin{equation*}
{\mathscr S}'(\mathbb{R}^{n}, H_0; H_1) := {\mathscr L}( {\mathscr S}(\mathbb{R}^{n}, H_0), H_1)
\end{...
3
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A question about Fourier transform of function of the type $Q(x)(1+P(x))^{z}$
For simplicity, consider in $\mathbb{R}^3$, and the Fourier transform of the following function
$$f=(x_1+x_2+x_3)(1+|x|^2+x_1^2(x_2^2+x_3^2)+x_2^2x_3^2)^{-t+is},~~ \frac12<t<1,~~s\in \mathbb{R}.$...
3
votes
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answers
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smoothing a current
Let $M$ be a smooth oriented manifold of dimension $n$ and $T$ a current of dimension $k$ on $M$. Let $\phi:P\times M \to M$ be a proper smooth family of diffeomorphisms of $M$ (i.e. $P$ is a smooth ...
3
votes
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262
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Fourier transform and support of a distribution
Let $T \in \mathscr D'(\mathbb R^n)$ be a distribution, such that its Fourier transform $\widehat T$ is a real analytic function on $\mathbb R^n$ but it can't be continued to an entire function on $\...
2
votes
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90
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Is the Schwartz space a tame Frechet space?
I ran into the following definition of tame Frechet spaces and Nash-Moser therem.
It says that the space of smooth functions on a compact manifold is tame Frechet.
However, I wonder if
The Schwartz ...
2
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Limit of a distribution using Hörmander’s theorem
Let $\alpha \in \mathbb{C}$. I want to prove that
$$ (e^{i2\theta}\xi_1^2 + \xi_2^2 + \dots + \xi_n^2)^{-\alpha} \longrightarrow (Q(\xi)-i0)^{-\alpha}, $$
in $D’(\mathbb{R}^n\setminus \left\{0\right\})...
2
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answers
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Blow up for certain classes of distributions
Let $\mathbb D$ be the open unit disc centered at the origin and let $u \in H^{-N}(\mathbb D)$ be a distribution for some natural number $N>0$. Suppose that $$u|_{\mathbb D\setminus \{0\}} \in C^{\...
2
votes
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answers
100
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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 ...
2
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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://...
2
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55
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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 ...
2
votes
0
answers
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Extreme confusion with the Gaussian measure on $\mathcal{S}'(\mathbb{R}^n)$ supported on $C^\infty(\mathbb{R}^n)$ and the issue of Borel sets
Let
\begin{equation}
C_a(x,y):=\frac{1}{(4\pi)^{n/2}} \int_a^\infty \frac{dk}{k^{n/2}}e^{-km^2-\lvert x-y \rvert ^2/(4k)}
\end{equation}
be a covariance operator with a cutoff $a>0$. Here, $m>0$ ...