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 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"...
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Given a compact set $K \subset \mathbb{R}^n$, is the space of distributions supported on $K$ the dual of some test function space?
I am aware that the dual of $C^\infty(\mathbb{R}^n)$ is the space of distributions (not necessarily tempered) with compact support.
However, if we fix a compact set $K \subset \mathbb{R}^n$, is the ...
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Defining the multiplication of distributions in the context of QFT : Colombeau algebra vs Regularity structure?
This is a bit of a qualitative question.
A rigorous treatment of QFT comes down to making sense of multiplication of distributions, as far as I understand. This is in the aim of constructing and ...
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What exactly is the topology on $O_M$ that makes the convolution map $S \times S' \to O_M$ hypocontinuous?
Let $O_M(\mathbb{R}^n):= \mathcal{S}'(\mathbb{R}^n) \cap C^\infty(\mathbb{R}^n)$ be the space of slowly increasing smooth functions on $\mathbb{R}^n$.
Following p.294 proposition 9.10 of the "...
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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, $m>0$ ...
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On spectral representation of solutions to wave equations with impulse initial data
Let $\Omega \subset \mathbb R^n$ be a bounded domain with a smooth boundary that contains the origin. Let us consider the following classical linear wave equation
$$
\begin{cases}
\partial^2_t u -\...
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Singular support: equivalent definition
Let $U\subset\mathbb{R}^{d}$ be an open set. The singular support of a distribution $u\in\mathcal{D}^{\prime}(U)$ is defined to be the compliment of the set of points, which have a neighbourhood in ...
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Representation of an operator on a generalized eigenfunction
This is a cross-post from: https://math.stackexchange.com/questions/4651664/representation-of-an-operator-on-a-generalized-eigenfunction
Suppose we have an (essentially) self adjoint operator $L$ ...
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Can we define $\partial\bar{\partial}(\log|z_1|^2)\wedge \partial\bar{\partial}(\log|z_2|^2)$ as a current?
In complex analysis, by Poincare-Lelong theorem, we have
$$
\frac{\sqrt{-1}}{\pi}\partial\bar{\partial}(\log|z|^2)=T_{z=0}
$$
as currents, where
$$
T_{z=0}(\eta)=\int_{z=0}\eta.
$$
Now suppose we have ...
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Newtonian potentials of balls and spheres
This is a simple question whose answer was probably known to Poisson, but I was not able to find it by searching. I need explicit formulas for the Newtonian potential of the unit ball $\mathbb{B}^n$ ...
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Subharmonic distributions on the plane
A subharmonic (Schwartz) distribution on $\mathbf R^n$ is a distribution $u$ satisfying $\Delta u\ge0$. This implies $\Delta u$ is a positive Radon measure $\mu$, thus for any ball $B$ the convolution ...
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Gauss's theorem under the convolution product
Assume that $\rho$ is a smooth scalar field in $\mathbf R^3$ and that $D$ is a measurable vector field in $\mathbf R^3$, such that, for every bounded domain $\Omega$ with smooth boundary $\partial \...
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Nuclear spaces and intuition behind their topology
In functional analysis the nuclear spaces (coined by Grothendieck before he became involved in revolutionizing algebraic geometry) can be considered
as a kind of generalization of finite dimensional ...
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Properties of the displacement field, assuming only smooth charge distribution and Gauss's theorem
In physics, the displacement field satisfies Gauss's theorem:
$$
\int_{\partial \Omega} {\bf D}\ {\bf n}\operatorname{d\!}S = \int_{\Omega} \rho\operatorname{d\!}V,
$$ where
$\Omega$ is a bounded ...
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Real-analytic analogue of Schwartz functions
Consider the space $\mathcal{S}'$ of functions $\mathbb{R}^n\to\mathbb C$ that are (real-)analytic and with exponential decay at infinity. This is an analogue of Schwartz space, but real-analytic ...
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Is there an example of a causally supported Schwartz function on $\mathbb{R}^4$ invariant under the Lorentz transform?
I am working on $\mathbb{R}^4$ with the sign convention $(1,-1,-1,-1)$.
I wonder if there is Schwartz function $f(x)$ on $\mathbb{R}^4$ such that the support satisfies the condition $0<x^2 < 4m^...
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Does Schwartz kernel theorem come from the universal property of tensor product?
In wikipedia we have Tensor product
The tensor product of two vector spaces $V$ and $W$ is a vector space denoted as $V \otimes W$, together with a bilinear map $\otimes:(v, w) \mapsto v \otimes w$ ...
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Confusion in notation of representation of Bastiani derivative
In the paper "Properties of field functionals and characterization of local functionals" at page 5 the Authors give the following definitions
Definition II.2. Let $U$ be an open subset of a ...
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Is there a general procedure to apply the regularity theorem for tempered distributions?
The regularity theorem for tempered distributions states that a tempered distribution is some weak derivative of a polynomially bounded continuous function.
For example, the delta function $\delta(x)$ ...
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Reference for Schwartz kernel theorem on vector bundles
In this notes Linear Analysis on Manifolds by Chris Kottke at page 20 he has
Theorem $1.16$ (Schwartz kernel theorem, c.f. [Hör85] Thm. 5.2.1). Let $M$ and $N$ be a compact Riemannian manifolds with ...
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How should I understand rigorously the definition of normal ordering of free fields
Let $\phi(x)$ be a free Hermitian scalar field in $4D$ Minkowski spacetime with the metric $(1,-1,-1,-1)$.
Then, though I wrote it as $\phi(x)$, it is in fact an operator-valued tempered distribution ...
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Generalizing a formula with distributions — Distributional Radon transform
I will try to describe the problem, it will necessarily be incomplete, so please if you have questions or remarks to make it more clear do not hesitate to leave them in comments.
The problem
Let $a$ ...
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Making sense of $1+1$ massless bosonic free field as a "distribution" rather than tempered
The question has been motivated by the fact that the $1+1$ massless bosonic free field suffers the infrared problem as a "tempered distribution".
The reason is essentially that $\int_{\...
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Problem In understanding distribuitional section
In this post Observables By Urs Schreiber he denotes the space of distributional sections in defenition 7.9 by $ \Gamma_{\Sigma}^{\prime}\left(E^*\right) $
That is if $u \in \Gamma_{\Sigma}^{\...
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Representation of the Dirac delta function
The Dirac delta function appears in the Sokhotsky formula,
$$\text{Im}\lim_{\epsilon\to 0^+} \frac{1}{x-i\epsilon} = \pi\delta(x),$$
to be understood in the integral sense
$$\text{Im}\lim_{\epsilon\to ...
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Does the (distributional) support of the Fourier transform of an $L^p$-function with $p<\infty$ have positive measure?
Suppose that $f \in L^p(\mathbb R^n)$ such that $1\leq p < \infty$. Let $\hat f$ be the Fourier transform of $f$. Clearly, if $p=1$ or $p=2$ then the support of $\hat f$ has positive Lebesgue ...
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Microlocal approach to definition of product of distributions
My question may be simple to an expert, but I'm not:
Let's consider $u \in C^{s}(\mathbb{R}^d)$ be a Hölder function sor some $s\in [0,1/2)$ which we may take very close to $0$.
Of course, $u^2 \in C^{...
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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 "...
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Propagators and PDEs
I have already asked this at MSE but did not get an answer.
In quantum field theory one encounters the retarded, advanced and Feynman propagators as certain solutions to a wave equation. ...
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Is it possible to extend Borel's lemma to the case of functional derivatives?
Let us think of a collection of tempered distributions $\{ T(x_1, \cdots, x_n)\}_{n=0}^\infty$. Here I will specifically set $x_i \in \mathbb{R}^4$ since I am considering quantum field theory and ...
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Chain rule in Sobolev space
In the theory of Sobolev space, we have the following chain rule:
For a uniformly Lipschitz function $F : \mathbf{R}\to \mathbf{R}$ such that $F(0)=0$,
and $u\in W^{1,1}(\mathbf{R}^n)$, then we have ...
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Schwarz space on the upper half-plane
Let us think of the Schwartz space $\mathcal{S}(\mathbb{R}^2_+)$ on the upper half-plane $\mathbb{R}^2_+=\mathbb{R}\times(0,+\infty)$ defined as
$$
\mathcal{S}(\mathbb{R}^2_+)=\left\{f\in C^\infty(\...
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Proof that elements of Beppo-Levi-like spaces are functions (and not just distributions)?
Context. I am trying to undestand the theory underlying "Beppo-Levi"-like spaces defined as
$$
H = \left\{f\in {\cal S}'(\mathbb{R}^d) \;\left| \; t\times\widetilde{f} \in {\cal L}^2(\mathbb{...
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A text about Schwartz distributions in vector bundles
If $M$ is a smooth manifold, one may talk about the space of test functions $\mathcal D (M)$ and its topological dual $\mathcal D ' (M)$ - the space of Schwartz distributions on $M$.
Now, if $E \to M$ ...
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How should I understand the "$C^\infty$ functions" whose domain is the dual of $C^\infty(\mathbb{R}^n)$?
I am reading Colombeau's book "New Generalized Functions and Multiplication of Distributions" and he uses the notation $C^\infty({C^\infty}'(\Omega))$ out of nowhere.
Here $\Omega$ is any ...
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What is the deep logic for the resonance function of dispersive nonlinear PDEs
I have been studying some nonlinear dispersive PDEs since some months and I was able to understand some results related to well-posedness. However, I do not feel like I am fully understand the logic ...
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Fourier transform of the hyperboloid
Equip $\mathbb{R}^{d+1}$ with the Lorentzian form $\langle x, y\rangle=-x^0y^0+{\bf x}\cdot{\bf y}$ where $x=(x^0,{\bf x})$ and $\cdot$ is the usual Euclidean dot product. We define the hyperboloid $\...
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Characterization of extendible distributions
I asked this question on Mathematics Stackexchange, but got no answer.
I found the following question which characterize the extension of a distribution in $\mathbb{R}$:
Let $f \in L_{\text{loc}}^{1}(...
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Request for a paper by Wightman and Gårding
I am looking for a pdf version of the following paper
A. Wightman and L. Gårding, Fields as operator-valued distributions in
relativistic quantum theory, Arkiv för Fysik 28 (1964), 129–189.
Does ...
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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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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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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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Distributions taking values in a TVS which is not locally convex
It seems to me that Schwartz's two seminal papers on vector-valued distributions only deals with distributions taking values in a locally convex Hausdorff topological vector space (LCS). Most other ...
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Pullbacks of LCS-valued distributions
Suppose $X$ is a locally convex space. Since the distributions $\mathcal{D}'\!(M)$ ($M$ a manifold) are a nuclear space, there is a canonical meaning to the topological tensor product $X\,\widehat{\...
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How should I understand the completeness relation of the form $\sum_{n} \phi_n(x) \overline{\phi_n}(y)=\delta(x-y)$?
Let $A$ be an unbounded self-adjoint operator on $L^2(\mathbb{R})$ and we are assuming the $L^2$ functions to be complex-valued.
We further assume (e.g. compactness of resolvent) that there exists an ...
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