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 ...
Gil Kalai's user avatar
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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
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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.
30 votes
6 answers
3k views

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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26 votes
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
25 votes
5 answers
3k views

Why are distributions "tempered"?

Google N-Gram shows that both "tempered distribution" and "temperate distribution" are used in English, but the first version significantly prevails, and usage of the second term ...
Alexandre Eremenko's user avatar
23 votes
6 answers
4k views

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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Eigenvectors of the Fourier transformation

The Fourier transform $\hat u$ is defined on the Schwartz space $\mathscr S(\mathbb R^n)$ by $ \hat u(\xi)=\int e^{-2iπ x\cdot \xi} u(x) dx. $ It is an isomorphism of $\mathscr S(\mathbb R^n)$ and the ...
Bazin's user avatar
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17 votes
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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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16 votes
2 answers
1k views

How to generalize the various vector calculus theorems to distributions?

Here is a list of vector calculus identities; in the proof of these identities, we all assume that these functions are $𝐶^𝑘$ in an open set, and we usually use these identities to calculate ...
YuerWu's user avatar
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16 votes
3 answers
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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 ...
Hugo's user avatar
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15 votes
2 answers
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Are Fourier transforms of L^p stable under diffeomorphisms?

Let $\xi$ be a compactly supported distribution on $\mathbb R^n$ and assume that its Fourier transform is in $L^p$. Let $\phi:\mathbb R^n\to\mathbb R^n$ be a diffeomorphism. Does the Fourier ...
Rami's user avatar
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14 votes
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
14 votes
1 answer
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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
14 votes
0 answers
663 views

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
1k views

Early successes of Schwartz distribution theory

What are the early successes of Schwartz distributions theory? What are the hard theorems that became simple and what open problems were solved with this new tool soon after Laurent Schwartz released ...
coudy's user avatar
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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
654 views

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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11 votes
2 answers
425 views

Trace on $\mathcal{S}(\mathbb{R}^k) \mathbin{\hat{\otimes}_\pi} \mathcal{S}'(\mathbb{R}^k)$

I asked this question on Math StackExchange, but it did not receive an answer, despite my offering a bounty to attract attention. I am unsure whether it is appropriate for this venue, but I thought ...
Matt Rosenzweig's user avatar
11 votes
2 answers
823 views

How do you know that you have succeeded-Constructive Quantum Field Theory and Lagrangian

Quantum Field Theory is a branch of mathematical physics which is begging for a better understanding. In fact there are no rigorous constructions of interacting QFT in four dimensions. By a rigorous ...
truebaran's user avatar
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10 votes
3 answers
3k views

Rationale for Hadamard's finite part of a divergent integral

(Note: I asked this question a few days ago on math.stackexchange but didn't get any responses. I've therefore decided to post it here instead.) I have a problem justifying throwing away the ...
Olumide'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
10 votes
1 answer
1k views

Distributions on product spaces

I hope this is suitable to MO. Question. Let $X$ and $Y$ be two open sets in $\mathbb R^n$ and $\mathbb R^m$, respectively. In what sense can we consider $\mathcal{D}^{\prime} \left(X\times Y\right)$ ...
darij grinberg's user avatar
10 votes
1 answer
401 views

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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10 votes
1 answer
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Research topics in microlocal analysis

Before asking this question here I did some research on web but I would like to get the opinion of those directly interested if there are any , (as I did in this thread Research topics in distribution ...
Andrew's user avatar
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10 votes
2 answers
1k views

Pull-back of generalized functions

Let $f\colon X\to Y$ be a smooth map between smooth manifolds. Then the pull-back operation $f^*\colon C^\infty(Y)\to C^\infty(X)$ is a linear continuous operator when $C^\infty$ is equipped with the ...
asv's user avatar
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9 votes
4 answers
803 views

Defining the value of a distribution at a point

Let $\omega\in D'(\mathbb R^n)$ be a distribution and $p\in \mathbb R^n$. If there is an open set $U\subset \mathbb R^n$ containing $p$ such that $\omega|_U$ is given by a continuous function $f\in C(...
B K's user avatar
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9 votes
1 answer
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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
9 votes
1 answer
871 views

Is the space of rapidly decreasing (non-smooth) functions nuclear?

We denote by $\mathcal{S}(\mathbb{R})$ the space of smooth and rapidly decreasing functions. We define on $\mathcal{S}(\mathbb{R})$ the family of semi-norms $$\lVert \varphi \lVert_{n,m} = \lVert (1+|...
Goulifet's user avatar
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9 votes
1 answer
2k views

Understanding Bruhat's notion of Schwartz function

I am trying to understand Bruhat's generalized Schwartz functions over (Hausforff) locally compact Abelian groups [1], following this paper [2] by Osborne. There, the Schwartz-Bruhat space $\mathscr{...
Juan Bermejo Vega's user avatar
9 votes
1 answer
377 views

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. ...
Bettina's user avatar
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9 votes
2 answers
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About the Fourier transform of the logarithm function

I want to calculate / simplify: $$\mathcal{F} (\ln(|x|)\mathcal{F(f)}(x))=\mathcal{F} (\ln(|x|)) \star f$$ where $\mathcal{F}$ is the Fourier transform ($\mathcal[f](\xi)=\int_{\mathbb R}f(x)e^{ix\...
Bertrand's user avatar
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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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8 votes
3 answers
610 views

Method to compute fundamental solutions which are distributions

The Malgrange-Ehrenpreis theorem tells us that there is a fundamental solution for any linear differential operator of constants coefficients. The original proof was not constructive (it was based on ...
Diego SolerPolo's user avatar
8 votes
1 answer
334 views

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
8 votes
1 answer
2k views

Topologies on spaces of distributions and test functions

Let $X$ be an open subset of $\mathbb{R}^n$. Following the notation of Schwartz, we denote $\mathcal{D}$ the space of compactly supported complex-valued smooth functions on $X$ equipped with the ...
Daps's user avatar
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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
8 votes
2 answers
465 views

Notion of generalized function/distribution for functional derivatives?

Is there any work on defining something analogous to a generalized function for functions whose domain is a Hilbert or Banach space? Is there an extension of the notion of Frechet/Hadamard/Gateaux ...
Will's user avatar
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8 votes
1 answer
381 views

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 \...
tsnao's user avatar
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7 votes
3 answers
1k views

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
393 views

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 ...
Isaac'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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7 votes
3 answers
2k views

Explicit isomorphism between distributions and universal enveloping algebra

The universal enveloping algebra of a Lie algebra $\mathfrak{g}$ is isomorphic to the algebra of distributions on the Lie group $G$ with support at the identity. The proof I have of this fact uses the ...
Najdorf's user avatar
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7 votes
1 answer
384 views

$\mathcal S(\mathbb R^n) \hat \otimes_\pi \mathcal S(\mathbb R^m) \simeq \mathcal S(\mathbb R^{n+m})$?

If $S(\mathbb R^n)$ is the Scwartz space of smooth rapidly decaying functions equipped with the topology generated by the family of semi-norms $$\mathcal N_p (\varphi)= \sum_{|\alpha|, |\beta| \leq p} ...
Thomas's user avatar
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7 votes
2 answers
409 views

On the Fourier-Laplace transform of compactly supported distributions

Let $\mathcal{E}'(\mathbb{R})$ be the space of all compactly supported distributions on $\mathbb{R}$. For $f\in \mathcal{E}'(\mathbb{R})$, let $\widehat{f}$ denote the entire extension of the ...
Giulia S-A.'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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7 votes
1 answer
501 views

On the definition of Hilbert spaces and real structures on Hilbert spaces

Let us consider the space $L^2:=L^2(\mathbb{R}^n,\mathbb{C})$ and the associated scalar product $S(f,g):=\int f \overline g$. In distribution theory, we have a situation where we have to deal with two ...
LCO's user avatar
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7 votes
1 answer
1k views

Regularity of Fourier transforms of $L^p$ functions for $2<p\le\infty$

I was recently reading about the Mikhlin and Hörmander Multiplier Theorems, which give conditions for a measurable function $m:\mathbb R^d\to\mathbb C$ to be an $L^p$ multiplier, i.e. for there to ...
Dominic Wynter's user avatar

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