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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Approximating a sequence of tempered distributions "uniformly" by Schwartz functions

This question has been motivated by the post making sense of distributions on the diagonal. Let $T$ be a tempered distribution on $\mathbb{R}^2$ and $\eta$ be a given mollifier on $\mathbb{R}$. For $f ...
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For a tempered distribution $F$ on $\mathbb{R}^2$, what exactly does it mean by $\lvert F(x,y) \rvert \leq \lvert x-y \rvert^{-n}$?

Let $F$ be a tempered distribution on $\mathbb{R}^2$ and $n \in \mathbb{N}$ be a fixed natural number. I wonder what exactly it means by $\lvert F(x,y) \rvert \leq \lvert x-y \rvert^{-n}$ where $x,y \...
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Construction of random tempered distributions

Let $(\xi_\phi)_{\phi \in L^2(\mathbb{R}_+ \times \mathbb{R}^d,\lambda_d)}$ be a collection of centered Gaussian processes on a probability space $(\Omega,\mathcal{F},P)$ such that $$\forall \phi \in ...
mathex's user avatar
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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 ...
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Topology of ${\mathcal D}(\Omega)$ (space of test functions)

I have seen two approaches to the topology of ${\mathcal D}(\Omega)$: (i) Let $K$ be a compact subset of $\Omega$ and consider the subset ${\mathcal D}_K(\Omega)$ of test functions with support ...
olih's user avatar
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Is it possible to bound the L2 norm of the gradient of a divergent by the L2 norm of the Lapacian?

Is it possible to show for $u:\Omega\subset\mathbb{R}^3\rightarrow \mathbb{R}^3$ that $$\||\nabla(\nabla\cdot u)|\|_2^2\leq C\||\Delta u|\|_2^2?$$ Here $\||f|\|_2$ is the norm in $(L^2(\Omega))^3$ and ...
Alberto Leandro's user avatar
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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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Proof that generalized Laplacian is essentially self-adjoint (Heat Kernels and Dirac Operators)

According to Proposition 2.33 in Heat Kernels and Dirac Operators each symmetric generalized Laplacian $H$ is essentially self-adjoint. This is an immediate consequence of the fact that \begin{...
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Finding an element of Gelfand triple with a designated time derivative

Let $V$ be a real separable Banach space and $H$ be a real separable Hilbert space such that \begin{equation} V \subset H \subset V' \end{equation} where $V'$ is the dual of $V$ and the inclusions are ...
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Special Schwartz function on the positive interval

Is there a Schwartz function $\zeta(t)$, defined on $\mathbb{R}$, satisfying the following: $\int \zeta(t)\: dt=1$, $\int t^k \zeta(t)\: dt=0$ for all $k\geq 1$, $\operatorname{supp}(\zeta)\subset (0,...
SnowRabbit's user avatar
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Examples of Borel probability measures on the Schwartz function space?

Let $\mathcal{S}(\mathbb{R}^d)$ be the Frechet space of Schwartz functions on $\mathbb{R}^n$. Its dual space $\mathcal{S}'(\mathbb{R}^d)$ is the space of tempered distributions. Minlos Theorem as ...
Isaac's user avatar
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Making sense of the limit $\lim\limits_{x \to y} T(x,y) $ for a tempered distribution $T$ on $\mathbb{R}^{2n}$

I already posted a similar question on MO and looked into the references therein. However, I cannot find a satisfactory answer for my question..So I ask here again in a more refined form. Let $T \in \...
Isaac's user avatar
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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^{\...
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Derive distributional inequalities from pointwise estimates

My question is how to prove the following claim: Suppose that $E$ is an algebraic set in $\mathbb{R}^n (n\ge3)$ with dimension $\le n-2$, and $u$ is locally Lipschitz continuous on $\mathbb{R}^n$. If ...
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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\})...
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Is this a well known space? Perhaps homogeneous Sobolev-like space?

The homogeneous Sobolev space $\dot H^s(\mathbb{R}^n) $ is often defined as the closure of $\mathcal{S}(\mathbb{R}^n)$ under the norm $$ || |\omega|^s \widehat{f} ||_{L^2(\mathbb{R}^d)} =\int_{\...
Dan1618's user avatar
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Non-Schwartz test functions for the explicit formula for L-functions

The statements of the explicit formula for L-functions that I am aware of require the test function to be a Schwartz function (see, e.g., equation (4.11) in Section 4 of Low lying zeros of families of ...
Tristan Phillips's user avatar
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Is there an infinite dimensional Stein's lemma?

Classical Stein's lemma says that if $\mathbf{X}$ is a centered Gaussian random vector and $g$ is a function which is nice enough, we have $$ \mathbb{E} \, X_i \, g ( \mathbf{X} ) = \sum_k \...
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A Gaussian measure $\mu$ on $\mathcal{E}'(S^1)$ by Minlos theorem and its value for Sobolev spaces $H^{\alpha}(S^1)$

I posted this question on ME as "A Gaussian measure on $\mathcal{E}'(S^1)$ by Minlos Theorem and its value for $L^2(S^1)$", but it seems much more nontrivial than I expected... so, I post an ...
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Infinite dimensional version of the Laplace transform and Gaussian integrals

This question is somehow related to my previous one 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) Borel-...
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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^{\...
Ali's user avatar
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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 ...
Isaac's user avatar
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Existence of adjoint operators on manifolds

Let $(M,g)$ be an oriented Riemannian manifold and $V$ a finite-rank vector bundle equipped with a non-degenerate bundle metric $\langle\cdot,\cdot\rangle_{V}$. This bundle metric, in turn, gives rise ...
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What real distributions solve $f'=0$? [closed]

I mean specifically real-valued Schwartz distributions on the real line.  That is linear functionals  on $C^{\infty}_c(\mathbb{R})$ continuous in the canonical LF topology.  My question is, what are ...
Colin McLarty's user avatar
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$K *g_n$ converges in the topology of smooth functions, $K$ approximates $\delta(x)$ and $g_n$ is a.e convergent to $g$, then regularity of $g$?

This question is continuation from If $K *g_n$ converges in the Fréchet topology of smooth functions and $K$ approximates $\delta(x)$, is $g_n$ itself convergent? - revised. As before, let us ...
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If $K *g_n$ converges in the Fréchet topology of smooth functions and $K$ approximates $\delta(x)$, is $g_n$ itself convergent? - revised

Let us consider the Fréchet space $C^\infty\Bigl([0,1],\mathbb{R} \Bigr)$ of real-valued, periodic smooth functions. That is, $f_n \to f$ in $C^\infty\Bigl([0,1],\mathbb{R} \Bigr)$ if $f^{(m)}_n$ ...
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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 ...
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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://...
fast_and_fourier's user avatar
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Banach space valued distributions and test functions

Let $A,B,C$ be Banach spaces and $m\,:\,A\times B\to C$ be a bilinear map such that $\|m(a,b)\|\leq \textrm{const}\,\|a\|\|b\|$. We denote by $\mathcal{S}(\mathbb{R}^d)$ be the standard space of ...
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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-...
Isaac's user avatar
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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 ...
Isaac's user avatar
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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 ...
Isaac's user avatar
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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 ...
백주상's user avatar
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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"...
Isaac's user avatar
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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 ...
Isaac's user avatar
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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 ...
Isaac's user avatar
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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 -\...
Ali's user avatar
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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 $L$ ...
APIs's user avatar
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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 ...
Zhaoting Wei's user avatar
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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 $\mathbb{B}^n$ ...
Piero D'Ancona's user avatar
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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 ...
Piero D'Ancona's user avatar
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1 answer
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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 ...
user267839's user avatar
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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 ...
MikeTeX's user avatar
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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 ...
Zislu R.'s user avatar
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1 answer
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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^...
Isaac's user avatar
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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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