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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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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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 ...
- 397
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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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answers
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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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votes
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answers
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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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Asymptotics of integral representation of distribution
I initially posted this question at MSE (here), but I have gotten no response, so I figured I would ask it to this community.
Background: I am studying the PDE $$\,\,\,\,\,\,\,\,\,\,\,\,i\partial_t \...
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Product of Heavisides: calculus vs Fourier transform vs wavefront set
I decided to ask this question here, since I did not get any answer from MSE and perhaps this topic is somewhat far from MSE's topics. I am following the paper here. I am trying to understand how to ...
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votes
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answer
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Fourier transform of a Radon measure [closed]
Let $\mu$ be a Radon measure on $\mathbb R^d$
with finite total mass: I guess that it is a tempered distribution on $\mathbb R^d$ and thus one may consider its Fourier transform. Now I guess that its ...
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Does the Ramanujan-Petersson condition correspond to a Fourier type property?
The Ramanujan-Petersson is one of the requirements used in Selberg's class of L-functions, and as such is a necessary condition for the Riemann Hypothesis to hold. The general converse conjecture ...
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de Rham theorem for tempered distributions
I am wondering if the following statement holds.
If $u\in \mathscr{S}'$ satisfies $\left< u,\Phi\right>=0$ for all $\Phi \in \mathscr{S}$ with $\mathrm{div}\, \Phi=0$, then there exists $p\in \...
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Looking for an electronic copy of Trèves and Martineau books
I would like to know if anyone has an electronic copy of the following books:
MR0177291 Martineau, A.; Trèves, F. Éléments de la théorie des espaces vectoriels topologiques et des distributions. Fasc....
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Radon transform of the function $h(x_1,\ldots,x_n) = x_1 g(x_1,\ldots,x_n)$, where $g$ is the density of multivariate Gaussian $N(\mu,\Sigma)$
Given an absolutely integrable function $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined for every $(w,b) \in (\mathbb R^n \setminus \{0\}) \times \mathbb R$ by
$$
R[f](w,b) := ...
- 6,726
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answer
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Restore initial condition for distributions
I am a bit confused by the following question and I hope someone could help me out.
Let $u$ be the solution of the following initial value problem
$$
u''(t) = g(t) \; \text{ in } (0,\infty), \quad\...
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vote
0
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Convergence in law and distribution theory
A standard result in probability theory asserts that a sequence of probablity measures $\mu_n$ on the Borel $\sigma$-algebra of $\bf R$ converges in law or weakly to a probability measure $\mu$ if and ...
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votes
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Functions of moderate increase compactly generated?
Let $\mathcal{O}_M(\mathbb{R}^d)$ be the space of smooth moderately increasing functions
$\{ f \in \mathcal{C}^\infty(\mathbb{R}^d) : \forall \alpha \exists N \text{ such that} \Vert \langle \cdot \...
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Real life applications of distributions through models or simulations [closed]
What are the areas we can apply distributions in classical harmonic analysis? I don't mean probability distributions but distributions that are continuous linear functionals on the space of test ...
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$\frac{\partial f}{\partial \overline{z}}=0$ in distributional sense implies $f$ is holomorphic
Let $f=(u,v)\in \mathscr{D}'(U,\mathbb{C})$ be a distribution, where $U\subset\mathbb{C}=\mathbb{R}^2$ is an open set and $u$ and $v$ are the projection of $f$ onto the real and imaginary axis (ie $\...
- 1,037
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votes
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Anti-delta function?
Did anyone ever consider a "function" or "distribution" $F(x)$ with the following property:
its integral $\int_a^b F(x)\,dx=0$ for any finite interval $(a,b)$ but $\int_{-\infty}^\...
- 9,312
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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-...
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