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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A roadmap to Hairer's theory for taming infinities

Background Martin Hairer gave recently some beautiful lectures in Israel on "taming infinities," namely on finding a mathematical theory that supports the highly successful computations from quantum ...
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5 answers
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Nice applications for Schwartz distributions

I am to teach a second year grad course in analysis with focus on Schwartz distributions. Among the core topics I intend to cover are: Some multilinear algebra including the Kernel Theorem and ...
Abdelmalek Abdesselam's user avatar
30 votes
6 answers
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Can distribution theory be developed Riemann-free?

I imagine most people who frequent MO have been indoctrinated into the point of view that the Riemann integral can be safely discarded once one has taken the time to develop the Lebesgue integral. ...
Paul Siegel's user avatar
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3 votes
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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 ...
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47 votes
9 answers
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What examples of distributions should I keep in mind?

I'm learning a bit about the theory of distributions. What examples of distributions will help me develop good intuition? Definitions: Let $U$ be an open subset of $\mathbb{R}^n$. Write $C_c^\infty(...
31 votes
19 answers
22k views

Good books on theory of distributions

Hi all. I'm looking for english books with a good coverage of distribution theory. I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions. Thanks in advance.
16 votes
3 answers
2k views

Why is multiplication on the space of smooth functions with compact support continuous?

I asked the question Why is multiplication on the space of smooth functions with compact support continuous? on M.SE sometime ago but I didn't receive a satisfactory answer. I was reading this ...
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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, ...
geocalc33's user avatar
8 votes
2 answers
981 views

Division of Distributions by Polynomials

Let $P(z)$ be a non-null complex polynomial in $n$ variables $z=(z_1,\dots,z_n)$: \begin{equation} P(z)=\sum_{|\alpha| \leq N} c_{\alpha} z^{\alpha}, \end{equation} where as usual for every $\alpha=(\...
Maurizio Barbato's user avatar
5 votes
1 answer
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Is every distribution a linear combination of Dirac deltas?

My question is whether Dirac-type distributions over an Abelian group define a basis of the Schwartz-Bruhat space $\mathcal{S}(G)^\times$ of tempered distributions on $G$, so that any distribution $f\...
Juan Bermejo Vega's user avatar
2 votes
2 answers
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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 $f^{(m)}_n$ ...
Isaac's user avatar
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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 ...
Isaac's user avatar
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23 votes
6 answers
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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}^\...
Anixx's user avatar
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17 votes
3 answers
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Research topics in distribution theory

The theory of distributions is very interesting, and I have noticed that it has many applications especially with regard to PDEs. But what are the research topics in this theory? also in terms of ...
Andrew's user avatar
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4 answers
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Topology on the space of Schwartz Distributions

If we equip the Schwartz space $\mathcal{S}$ with its usual Fréchet space topology, then the space of continuous linear functionals $\mathcal{S}^\ast$ is known as the space of Schwartz distributions ...
Jonathan Gleason's user avatar
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0 answers
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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 "...
Chris Wendl's user avatar
12 votes
1 answer
638 views

Wavelet-like Schauder basis for standard spaces of test functions?

Edit: A more precise formulation of my question follows the separation line. The Schwartz space of test functions $\mathcal{S}(\mathbb{R})$ is isomorphic to $\mathfrak{s}$ the space of sequences of ...
Abdelmalek Abdesselam's user avatar
12 votes
1 answer
880 views

On an Inequality of Lars Hörmander

Let $P(z)$ be a non-null complex polynomial in $\nu$ variables $z=(z_1,\dots,z_n)$ of degree $\mu$: \begin{equation} P(z)=\sum_{|\alpha| \leq \mu} c_{\alpha} z^{\alpha}, \end{equation} where as usual ...
Maurizio Barbato's user avatar
11 votes
2 answers
685 views

Poincaré lemma for distributions

Let us consider a current on $\mathbb R^n$, that is a differential form whose coefficients are distributions. For simplicity, let us check the case of a $1$-form $$ u=\sum_{1\le j\le n} u_j dx_j,\quad ...
Bazin's user avatar
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11 votes
1 answer
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Is every continuous endomorphism of the Schwartz space a pseudo-differential operator?

Let $\mathcal{S}:= \mathcal{S}(\mathbb{R}^n)$ be the Schwartz space of smooth functions with rapid decay. The question is pretty simply stated in the title. Pseudo-differential act continuously on the ...
Saal Hardali's user avatar
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10 votes
1 answer
639 views

Are functions of moderate growth a bornological space?

I was thinking a bit about distribution theory the last weeks and stumbled across the following question: There are two natural locally convex topologies on the space of smooth functions of moderate ...
Johannes Hahn's user avatar
9 votes
1 answer
4k views

Fourier transform of a bounded function

This should really be well-known, but I was not able to find a definite answer to this question: Is the Fourier transform of a bounded function always a borel measure (i.e. an order 0 distribution)? ...
Matthias Ludewig's user avatar
8 votes
6 answers
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Fourier transform of (real) exponential

Is it possible to make sense, in distributional sense, of the Fourier transform of the exponential function (defined over the whole real line)?
johny's user avatar
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3 answers
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The "Spaces of Schwartz distributions are finite dimensional" challenge

The more I study Schwartz distributions and the corresponding spaces, the more the latter look "finite dimensional" to me. Of course they are not finite dimensional in the technical sense but they are ...
Abdelmalek Abdesselam's user avatar
7 votes
2 answers
440 views

Distribution that vanishes against approximated delta is zero

Suppose we have a Schwartz distribution $\phi$ on $\mathbb{R}^d$ such that $$ \forall x, \ \lim_{\lambda \to 0}| \langle\phi, \psi^{\lambda}_x \rangle| =0$$ where $\psi^{\lambda}_{x}=\lambda^{-d}{\...
Kore-N's user avatar
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1 answer
189 views

$GL_1(\mathcal{E}'(\mathbb{R}))$ open in $\mathcal{E}'(\mathbb{R})$?

Let $\mathcal{E}'(\mathbb{R})$ be algebra of all compactly supported distributions on $\mathbb{R}$, equipped with the strong dual topology $\beta(\mathcal{E}',\mathcal{E})$, and with the usual ...
Giulia's user avatar
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7 votes
2 answers
954 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 ...
iolo's user avatar
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0 answers
496 views

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[\...
Abdelmalek Abdesselam's user avatar
6 votes
1 answer
795 views

Composition of a smoothing operator with an $L^2$-bounded operator, non-compact Riemannian manifold

I'm trying to close in on a definitive answer to my own question BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?, and think I ...
JahvedM's user avatar
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1 answer
839 views

Solve $f(x)=\int_{x-1}^{x+1} f(t) \mathrm{d}t$

Solve $f(x)=\int_{x-1}^{x+1} f(t) \mathrm{d}t$. When $f$ is a function, it looks like the only solution is $f(x)=0$. But what if we allow distributions, such as the Dirac delta?
Ricbit's user avatar
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1 answer
389 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 ...
user449595's user avatar
6 votes
0 answers
157 views

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) := \...
alvarezpaiva's user avatar
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5 votes
1 answer
440 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 & ...
genfuntranslate's user avatar
5 votes
0 answers
233 views

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,\...
Asaf Shachar's user avatar
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5 votes
2 answers
523 views

$\frac{\partial f}{\partial \overline{z}}=0$ 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 $\...
No-one's user avatar
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4 votes
1 answer
260 views

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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4 votes
0 answers
103 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 ...
Kosh's user avatar
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4 votes
1 answer
321 views

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 ...
Isaac's user avatar
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3 votes
1 answer
215 views

Mathematical meaning for the (continuous) Sine-Gordon transformation

I've been trying to understand the so-called Sine-Gordon Transformation which occurs in both classical and quantum statistical mechanics. One of the most cited references on this topic seems to be ...
IamWill's user avatar
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2 votes
1 answer
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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 ...
Isaac's user avatar
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2 votes
0 answers
84 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 ...
Math604's user avatar
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1 vote
2 answers
246 views

Continuous Linear Mappings on Subspaces of $\mathcal{D}(\Omega)$

Let $\Omega$ be a non-empty open subset of $\mathbb{R}^n$ and $\mathcal{D}(\Omega)$ the usual space of test functions of distribution theory, with the usual topology $\tau$ of the inductive limit of ...
Maurizio Barbato's user avatar
0 votes
1 answer
657 views

($n$-dimensional) Inverse Fourier transform of $\frac{1}{\| \mathbf{\omega} \|^{2\alpha}}$

Note: I first posted question on math.stackexchange and I got one reply, which was a bit helpful (I'm still trying to understand it fully), but did not explore the two solution cases that I mentioned. ...
Olumide's user avatar
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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 \...
Isaac's user avatar
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0 votes
1 answer
145 views

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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1 answer
690 views

The dual space of the Dirac measures on an Abelian group

Let $G$ be a Hausdorff locally-compact Abelian group and $L^2(G)$ the Hilbert space of two-integrable complex functions on the group. Question. What would be natural vector space $\mathcal{R}$ of ...
Juan Bermejo Vega's user avatar