**7**

votes

**1**answer

416 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+|...

**1**

vote

**1**answer

87 views

### Estimate the analytical wavefront set $WF_A(u)$ given $WF_A(A_K u)$

Let $X$ and $Y$ be two real-analytic manifolds and $Z \subset X \times Y$ be a real-analytic embedded closed submanifold. Suppose now that $K \in \mathscr D'(X \times Y)$ is a distribution conormal to ...

**5**

votes

**1**answer

610 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?

**4**

votes

**1**answer

200 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 \...

**0**

votes

**1**answer

268 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 $\mathcal{R}$ ...

**3**

votes

**0**answers

159 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 $\...

**2**

votes

**1**answer

471 views

### 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\...

**2**

votes

**1**answer

410 views

### Is the space of test functions separable? [closed]

Consider the space $\mathcal D(\mathbb{R}^n)$ of smooth functions (in the sense of having continuous derivatives of all orders) which are compactly supported. Endow it with its usual topology, i.e., ...

**3**

votes

**0**answers

232 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.

**4**

votes

**1**answer

553 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{...

**1**

vote

**1**answer

280 views

### Fubini for distributions which are not measures?

We have a "nonnegative" distribution $\mu$ with compact support in $\mathbb{R}^2$ which is not a measure, as we can produce a linear function $f(x,y)=x-1$ such that the integral of $f^{2k}$ w.r.t. $\...

**3**

votes

**1**answer

274 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, ...

**9**

votes

**2**answers

542 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 ...

**4**

votes

**0**answers

108 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 ...

**1**

vote

**1**answer

130 views

### countably normed spaces and countably normed spaces [closed]

Why locally convex spaces are not presented as countably normed spaces i.e an infinite sequence of norms (see Generalized functions Tome 2 by Gelfand and Chilov) in the western mathematical ...

**0**

votes

**1**answer

746 views

### Dirac delta composed with absolute value [closed]

I hope this question is well suited for this site; please excuse me if not.
I recently read that the value of $\delta(x^2)$ is an open question [1], with $\delta(x)$ the Dirac delta. Now I'm trying ...

**0**

votes

**1**answer

136 views

### Why equality of singular supports?

Let $f:\mathbb{R}^n\to \mathbb{R}$ be a smooth and bounded function which tends to 0 at infinity. Define, for $t>0$, the distribution
$$
\nu (t) = \int \limits _{f(x)\ge t} dx,
$$
and (in the ...

**1**

vote

**1**answer

246 views

### Support of a distribution

Let $g:\mathbb{R}^n \to \mathbb{R}$ be a smooth real-valued function that decays like some positive power of $|x|^{-1}$ at infinity. Define the first order distribution $\mu $ by
$$
\langle \mu , \...

**3**

votes

**1**answer

326 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 $t_+=\max (t,0)$.)

**2**

votes

**0**answers

701 views

### Nonlinear PDE and Green functions

This is somewhat of a curiosity that can hide somewhat deeper. For a Green function of a nonlinear PDE I mean something like
$$
\partial^2\phi+V(\phi)=\delta^D(x).
$$
I do not know if a real ...

**3**

votes

**1**answer

982 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)?
...

**0**

votes

**3**answers

413 views

### Regularized fractional derivative of distributions.

A fractional derivative of distributions is usually introduced using definition of fractional integral as a convolution of two distributions. However there is another approach based on fractional ...

**2**

votes

**2**answers

140 views

### Inverse schwartz-distribution for convolution operation

I note here $\mathcal{D}'$ the space of all distributions and $\mathcal{S}'$ the space of tempered distributions, I am considering the following question:
Let $u \in \mathcal{D}'$ or $\mathcal{S}'$, ...

**3**

votes

**2**answers

568 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 ...

**2**

votes

**1**answer

159 views

### Distributional limits concerning the regularity of Maxwells equations

This question is related to my previous question about the regularity of the Maxwell equations.
Assume we are working on a space where there are only electric point charges, $(q_i)$, and a blob of ...

**4**

votes

**4**answers

432 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", "...

**0**

votes

**2**answers

375 views

### Eigenfunction of local fractional derivative

Let $E_{\alpha}(x^{\alpha})$ be a Mittag-Leffler function, $\alpha \in (0,1)$. It is an eigenfunction for nonlocal fractional derivative, defined as a convolution with
$$
\Phi_{\lambda}(x) = \frac{...

**2**

votes

**2**answers

448 views

### Fourier transform of $e^{it|\xi|^{\alpha}}$

Consider the fourier transform of $e^{it|\xi|^{2\alpha}}$ ($\alpha>0$)in $\mathbb{R}^n$,let $K_{\alpha}=\mathcal{F}(e^{it|\xi|^{\alpha}})$,so $K$ is a tempered distribution.Now I want to know if ...

**4**

votes

**1**answer

706 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 $\...

**5**

votes

**0**answers

331 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 ...

**0**

votes

**1**answer

426 views

### Calculating a distributional derivative

Suppose that we have a sequence of functions $u_j$ that are in $L^{\infty}(0,1)$. Then the sequence of maps $N_j(s) := \|u_j(s)\|^2$ are also in $L^{\infty}(0,1)$. Hence they give rise to ...

**1**

vote

**2**answers

483 views

### Opinions on the Multiplication of Measures

A few questions, hopefully to spark some discussion.
How can one define a product of measures?
We could use Colombeau products by embedding the measures into the distributions? I'm not sure why ...

**3**

votes

**2**answers

466 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 ...

**5**

votes

**2**answers

283 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 \mathcal{E}(\Omega)...

**0**

votes

**1**answer

1k views

### The fourier transform of homogeneous distribution and related topics

When we have a distribuion $u\in \mathcal{D}'(R^n)$,and the restriction to $R^{n}\backslash{0}$ is homogeneous of degree a,we have $u \in \varphi'$ and $\widehat u$ is of degree(-n-a) in $R^{n}\...

**2**

votes

**3**answers

879 views

### The topology of $C_0^\infty(M) $

I have read definitions in my PDE book as follows: If $M$ is a smooth paracompact manifold, the space of all linear functional on $C^\infty(M)$ is denoted by $E'$ and the space of all linear ...

**0**

votes

**2**answers

806 views

### Is there dual space of the distributions $\mathcal{D}'(R)$?

Dear MOs,
Let $\mathcal{D}(R):=C_c^\infty(R)$ be the smooth functions with compact support. Its dual space is the space $\mathcal{D}'(R)$ of distributions. This space $\mathcal{D}(R)$ has its weak *-...

**2**

votes

**0**answers

182 views

### Spectral gap of tempered distributions

Hi,
Let $\Lambda\subset\mathbb{R}$ be an infinite discrete set of finite density (for simplicity one may take the density equals 1) and $\delta_{\lambda}$
is a unit mass located at the point $\lambda\...

**4**

votes

**2**answers

640 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, $...

**10**

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 ...

**4**

votes

**2**answers

469 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 ...

**2**

votes

**1**answer

425 views

### about decomposition of a non-negative definite operators

Hello,
Many years before, I had the following problem.
We first give a definition. Given a non-negative definite real-valued definite matrix $n^2\times n^2$ matrix $M$, it is called separable if it ...

**6**

votes

**3**answers

796 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. ...

**1**

vote

**1**answer

420 views

### Equivalent references for Schwartz's book of the distribution theory

Hello,
It seems that there is no English translation of the Schwartz's book 1966. I may need to use the spaces like
$$
\dot{\mathcal{B}}(R),\quad \dot{\mathcal{B}}'(R),\quad \mathcal{B}(R),\quad \...

**1**

vote

**1**answer

460 views

### A question about an equivalent definition of the Schwartz distribution

Hello,
Does anyone know a reference or proof of the "if" part of the following statement?
$$
\mu\in \mathcal{S}'(R)\quad\text{if and only if}\quad \mu*\alpha\in\mathcal{S}(R),\forall \alpha\in C_c^\...

**4**

votes

**2**answers

1k 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 ...

**0**

votes

**1**answer

422 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. ...

**6**

votes

**1**answer

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

**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 ...

**3**

votes

**1**answer

470 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 ...