**24**

votes

**7**answers

2k views

### 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 ...

**12**

votes

**18**answers

7k 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.

**9**

votes

**2**answers

405 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 ...

**8**

votes

**4**answers

2k views

### 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 ...

**8**

votes

**2**answers

343 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 ...

**7**

votes

**3**answers

1k 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 ...

**7**

votes

**1**answer

312 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 ...

**6**

votes

**1**answer

610 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 $\mathcal{D}^{\prime} \left(X\times Y\right)$ ...

**6**

votes

**0**answers

122 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) := ...

**5**

votes

**4**answers

1k views

### Distributions more complicated than the Dirac δ and derivatives

The responses to another question clarifies that the best known examples of distributions that are not measures, are the derivatives of the delta and such. What I want to know is: Is that the only way ...

**5**

votes

**1**answer

579 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?

**5**

votes

**3**answers

729 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 ...

**5**

votes

**1**answer

171 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 ...

**5**

votes

**3**answers

739 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. ...

**5**

votes

**1**answer

185 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} ...

**5**

votes

**3**answers

193 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 ...

**5**

votes

**2**answers

275 views

### Integration under functional sign

Let $f(x,y)$ be some bounded with its derivatives continuous function on $\Omega \times \overline{\Omega}$, where $\Omega$ is a domain in $\mathbb{R}^n$. Let $f(\,\,\cdot\,,\,y) \in ...

**5**

votes

**2**answers

217 views

### Eigenstates of Fourier transformation

Let $\gamma$ be defined on $\mathbb R^n$ by $\gamma (x)=e^{-π x^2}$. With $\mathcal F$ standing for the Fourier transformation defined on the Schwartz space by
$$
(\mathcal F u)(\xi)=\int e^{-2iπ ...

**5**

votes

**0**answers

81 views

### Support of a Measure with Characteristic Functional Continuous in $L_p$, $1\leq p <2$?

Let $\mathcal{S}(\mathbb{R})$ be the space of smooth and rapidly decaying functions and $\mathcal{S}'(\mathbb{R})$ its dual, the space of tempered distributions. Let $\mu$ be a probability measure ...

**4**

votes

**4**answers

373 views

### Reference for integral of functions taking values in a topological vector space.

(Note that I am interested in the Gelfand-Pettis integral specifically, as opposed to, for example, the Bochner integral.) I have tried Googling things like "integral topological vector space", ...

**4**

votes

**2**answers

490 views

### Wightman fields vs local functionals vs operators

In QFT literature one wants to look at $n-$point correlation functions of "operators" inserted at $x$ say, $\cal{O}(x)$ and if $\phi_i$ are the fields then the quantity one has in mind is written as, ...

**4**

votes

**2**answers

427 views

### Decomposition of distributions

Can we write every (tempered) distribution $\psi$, say on $\mathbb{R}$, as the sum of two distributions
$\psi = \psi_1 + \psi_2$
such that $\psi_1$ and the Fourier transform of $\psi_2$ are ...

**4**

votes

**1**answer

152 views

### Practical way to check whether a distribution is conormal

Let $X$ be a $C^\infty$-manifold, $Y$ be its $C^\infty$-submanifold. Hörmander defines the set $I^m(X,Y)$ of conormal distributions as the set of all $u \in \mathscr D'(X)$ such that
$$
L_1 ...

**4**

votes

**1**answer

221 views

### Calculus of variation

This is probably simple but I'm stuck somewhere. I am trying to solve the calculus of variation problem that arise in an applied field: $$\min_{f \in C^1} \int^1_0 \int^1_0 (x-y)^2f(x,y)dxdy$$ ...

**4**

votes

**1**answer

552 views

### What functions can be obtained as a convolution of a Schwartz function and a tempered distribution?

Let $\mathcal S (\mathbb R)$ denote the space of Schwartz functions on $\mathbb R$ and $\mathcal S^* (\mathbb R)$ denote the dual space of Schwartz (a.k.a tempered) distributions.
We consider ...

**4**

votes

**0**answers

147 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)}
...

**4**

votes

**0**answers

118 views

### Define the space of distributions with algebraic decay?

A tempered distribution $u\in \mathcal{S}'(\mathbb{R})$ is said to be rapidly decreasing if for every $f \in \mathcal{S}(\mathbb{R})$, $u*f \in \mathcal{S}(\mathbb{R})$.
One rough way to motivate ...

**4**

votes

**0**answers

122 views

### Convergence of Schwartz Kernels

I read this question, and I would like to ask the opposite: Assume that I have a sequence of smoothing operators $(P_n)$ with (hence smooth) kernels $(p_n)$ converging strongly to some smoothing ...

**4**

votes

**0**answers

261 views

### distributions on Lie groups and representations

Let $G$ be a Lie group and $\pi$ a continuous action on $V$, a Fréchet space.
This action induces a representation of the space of compactly supported functions, $C_c(G)$, with convolution as product ...

**3**

votes

**2**answers

914 views

### Distributions and measures

Hello,
After reading the previous post, I still have some doubts. Let's consider everything on $R$ to avoid complications.
Can we say that any distribution $\mu\in\mathcal{D}'(R)$ of zero order ...

**3**

votes

**3**answers

265 views

### When sequentially continuous linear functional is continuous?

Let $C^\infty(X)$ denote the space of infinitely smooth functions on a compact manifold $X$ (at the beginning one may assume that $X$ is a circle, though I need a more general case). Let ...

**3**

votes

**1**answer

618 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)?
...

**3**

votes

**2**answers

435 views

### Schwartz kernel theorem for topological spaces

Is there some regularizing version of Schwartz kernel theorem for topological spaces, i.e., in the form of
Every continuous linear map $A\colon C\prime(X_2) \to C(X_1)$ is given by a kernel $k \in ...

**3**

votes

**1**answer

1k views

### A good reference for the wave front set

Hello,
I am wondering whether anyone know some good references for the theory of wave front set, microlocal analysis? I have some basic knowledge of distribution theory at the level of the Rudin's ...

**3**

votes

**1**answer

297 views

### Fourier transform of tempered distribution

I'm wondering whether anyone knows a reference or proof for finding the Fourier transform of $f(t):=(t+1)^{1/2}t_+^{1/2}$? (Here $t_+=\max (t,0)$.)

**3**

votes

**1**answer

233 views

### Can I approximate Schwartz functions which integrate to zero by $C_0^\infty$ functions which integrate to zero?

Let $X$ be the closed subspace of Schwartz space $\mathcal{S}(\mathbb{R}^N)$ defined by
\begin{equation*}
X=\left\{f\in\mathcal{S}(\mathbb{R}^N):\quad \int f\; dx=0\right\}.
\end{equation*}
My ...

**3**

votes

**2**answers

168 views

### Linear operators on distributions with different topologies

Denote by $\mathscr{D}^\prime$ and $\mathscr{D}^\prime_b$ the space of distributions on $\mathbb{R}^n$ equipped with the weak and the strong topology, respectively. Because the topology of ...

**3**

votes

**1**answer

250 views

### How to define a generalized differential form through its values on submanifolds

Suppose we're in $\mathbb R^n$, and we have a function on line segments ,$\omega(I)$, with values in $\mathbb R$. Give sufficient conditions for $\omega$ to be given by a generalized 1-form (that is, ...

**3**

votes

**2**answers

424 views

### One-sided Cauchy principal value

What is the notion of a principal value of an integral when the singularity appears at one endpoint? Namely,
$ PV \int_a^b f(t) dt = ? $,
where the integral is convergent in the upper limit, but ...

**3**

votes

**1**answer

440 views

### Ergodicity of Convoluted White Noise

I have a question regarding ergodicity in infinite dimensional spaces.
Let $\mathcal{D}$ be the space of distributions on a Schwartz space, and let $\mu$ be the white noise process which exists by ...

**3**

votes

**1**answer

120 views

### Zero currents localized along a submanifold

Let $\mathcal{D}(\mathbb{R})$ be the continuous dual of $C^\infty_c(\mathbb{R})$, the space of compactly-supported smooth functions. There is a nice characterization of distributions ...

**3**

votes

**1**answer

356 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 ...

**3**

votes

**1**answer

163 views

### Extension of pseudodifferential operators

I'm very sorry if this is the wrong place to ask this question, but I've asked it on StackExchange and received no answers. ( ...

**3**

votes

**1**answer

591 views

### An elementary introduction of Colombeau's generalized function theory

Hello, I am wondering whether anyone know an elementary reference for Colombeau's theory on the multiplication of distributions? I encountered the problem of the square of Delta function. I need a ...

**3**

votes

**1**answer

1k views

### Existence of weak limits

Background: Three months ago, I asked this question, which is a bit related to the following (if the answer to it was Yes, then answer to this one would be Yes too, but since that was a No, it still ...

**3**

votes

**0**answers

153 views

### smoothing a current

Let $M$ be a smooth oriented manifold of dimension $n$ and $T$ a current of dimension $k$ on $M$. Let $\phi:P\times M \to M$ be a proper smooth family of diffeomorphisms of $M$ (i.e. $P$ is a smooth ...

**3**

votes

**0**answers

144 views

### Fourier transform and support of a distribution

Let $T \in \mathscr D'(\mathbb R^n)$ be a distribution, such that its Fourier transform $\widehat T$ is a real analytic function on $\mathbb R^n$ but it can't be continued to an entire function on ...

**3**

votes

**0**answers

211 views

### Why distributions as functionals? [closed]

Why do we generalize functions by functionals on Schwartz Spaces, beyond the fact that it simply works? There should be a deeper reason why Schwartz considered functionals.
Excited for answers, Alex.

**3**

votes

**0**answers

101 views

### Is there an analogue of distributions in characteristic p?

Some motivation: When working over $\mathbb{C}$, distributions (in the sense of generalized functions) act as natural generators for $D$-modules (in the sense that any regular holonomic $D$-module is ...

**2**

votes

**5**answers

2k views

### 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)?