Use this tag for questions about Schwartz distributions, also known as Generalised Functions. For questions about "probability distributions", use (probability-distributions). For questions about distributions as sub-bundles of a vector bundle, use (differential-geometry).

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7
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111 views

In the space of distributions, how big is the subspace of functions?

I'm teaching Distribution theory and many of my students still believes that there is only one kind of distribution : the distribution that can be identified to a $L^1_{\text{loc}}$ function. And I ...
7
votes
0answers
284 views

Connection between distributional and renormalized solutions for Boltzmann equation

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read through some papers by DiPerna and Lions concerning the Cauchy Problem ...
6
votes
0answers
135 views

Two possible definitions of “vector-valued distribution”

Let $X$ be a reflexive Banach space. Define $$\tag{1} \mathcal{D}^\star(0, T; X)=\left\{ u\colon \mathcal{D}(0, T)\to X\ \text{linear and continuous}\right\} $$ where the topology on the space of ...
5
votes
0answers
55 views

Why do we give $C_c^\infty(\mathbb{R}^d)$ the topology induced by all good seminorms?

Briefly, my question boils down to the following: What benefits do we gain from considering the space of test functions in the topology induced by all "good" seminorms, as opposed to other topologies ...
5
votes
0answers
154 views

Isomorphism between $C^\infty_0(B_1)$ and $\mathscr{S}(\mathbb{R}^n)$

Background: Related question I am trying to prove, that the countably-normed spaces $C^\infty_0(B_1)$ on the open unit ball (i.e. function and all derivatives vanish at the boundary) in ...
5
votes
0answers
274 views

If a Radon measure is a tempered distribution, does it integrate all Schwartz functions?

The question might at first sight sound like the answer is trivially "yes", so let me clarify the question a bit. Consider given a Radon measure $\mu$ on $\mathbb{R}^n$. Let ...
5
votes
0answers
91 views

Show that integrals are equal

Let $f \colon [0,+\infty) \to \mathbb R$ be a convex function. Then $f''(x)$ is a nonnegative distribution on $(0,+\infty)$ and hence it can be continued to a nonnegative mesure $\mu$ on ...
4
votes
0answers
28 views

Rigorous Justification of Infinitesimal Techniques

As you may know that there are a bunch of heuristic techniques in physics to make integrals converge. For example, when we define a following Fourier transform, we add a positive infinitesimal and let ...
4
votes
0answers
72 views

How to prove that this solution of heat equation is not a tempered distribution?

A theorem of PDEs sais that the following Cauchy problem for the heat equation \begin{align*} & \partial_t u = \partial_{x}^2 u, \quad (t,x) \in \mathbb{R_+} \times \mathbb{R}, \\ & u|_{ t = ...
4
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0answers
33 views

Distributions and primitives

I was wondering: if distributions are seen as a generalization of functions that "removes obstructions" to the operation of derivation, is there a generalization of functions that would remove any ...
4
votes
0answers
72 views

Limit of distribution

Let $T\in\mathcal{D}'(\mathbb{R})$ be a distribution on the set of smooth functions of compact support $\mathcal{D}(\mathbb{R})$ such that $$ \forall_{g\in\mathcal{D}(\mathbb{R})}~|\langle T, g ...
4
votes
0answers
78 views

why is test function space $\mathcal{A}$ complete

I am trying to find out, why the space $$\mathcal{A}:=\left\{\phi\in C_0(\mathbb{R}^{2d})|\;\|\phi\|_\mathcal{A}:=\int_{\mathbb{R}^d}\sup_{x\in\mathbb{R}^d}|(\mathcal{F}_p\phi)(x,y)|\;\mathrm ...
4
votes
0answers
94 views

Regularity theorem for Laplacian

Let $\Omega \subset \mathbb R^d$ be a bounded domain, $d>2$. Let $f \in C^\infty(\Omega)$. If $u \in L^2$ is a distributional solution of $\Delta u = f$ in $\Omega$ then $u \in C^\infty(\Omega)$ ...
4
votes
0answers
149 views

Distributional derivative of bounded functions

Let $f$ be a bounded measurable function on $\mathbb{R}$. We may consider its derivative $f'$ as a distribution on $\mathbb{R}$. Is there a reasonable description of those distributions $\psi$ ...
3
votes
0answers
35 views

Distribution induced by a function

We are given $F(x) = |2x+1|, x \in \mathbb{R}$ How to determine whether $$[F|_{(0, \infty)}] \in \mathcal{D}'((0, \infty))$$ $$[F|_{(- \infty, 0)}] \in \mathcal{D}'((- \infty, 0))$$ are regular ...
3
votes
0answers
74 views

Klein-Gordon field commutator integral?

Consider a Klein-Gordon field $\phi$, which satisfies $$(\Box+ \omega_0^2)\phi=0$$ on points $x \equiv \{x_0,\vec{x}\},y\equiv \{y_0,\vec{y} \}$ of 4D Minkowski-spacetime. The field commutator is $$ ...
3
votes
0answers
32 views

What is the topological interpretation of continuity of distributions?

I was given this definition of continuity in the distributional sense. A distribution $T$ over the space of test functions $\mathcal{D}$ is continuous if for every sequence of test functions $\{ ...
3
votes
0answers
62 views

What is a good way to prove the weak form of the Sobolev embedding theorem?

From Richard Melrose, From microlocal to global analysis, Chapter 1, Problem 11. Suppose $u\in L^{2}(\mathbb{R}^{n})$ and that $$ D_{1}D_{2}\cdots D_{n}\mu\in (1+|x|)^{-n-1}L^{2}(\mathbb{R}^{n}) $$ ...
3
votes
0answers
68 views

Hopf lemma for generalized normal derivatives

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. If $u\in C^2(\Omega)\cap C_0(\overline{\Omega})$ is a superharmonic function ($-\Delta u\ge 0)$ then, Hopf Boundary Lemma does implies that ...
3
votes
0answers
51 views

Do we have $C^\infty \cap \mathcal{O}_C' = \mathcal{S}$ and/or $C^\infty \cap \mathcal{S}' = \mathcal{O}_M$?

We define the following traditional function spaces from distribution theory. $\mathcal{S}$ the space of rapidly decreasing smooth functions. $\mathcal{S}'$ the space of tempered distributions, dual ...
3
votes
0answers
70 views

Exponential of the derivative operator on the Schwartz space?

We consider the derivative operator $\mathrm{D}$ on the space of smooth and rapidly decreasing function $\mathcal{S}$. We denote by $P_n = \frac{1}{0!} + \frac{X}{1!} + \frac{X^2}{2!} + \cdots + ...
3
votes
0answers
123 views

Verifying the fundamental solution of Stokes flow in R^3?

Recall that Stokes flow in $\mathbb{R}^3$ is the solution $(\textbf{u}, p)$ of $$\begin{align} -\nabla p + \mu \Delta \textbf{u} &= \textbf{f} \\ \nabla \cdot \textbf{u} &= 0, \end{align} ...
3
votes
0answers
101 views

Variation of Partition of Unity

We know as "Partition of unity" that follow: Let $X\subseteq \mathbb{R}^n$ be an open set, and let $K$ be a compact subset of $X$. Let $X_i$, $i=1,\ldots, m$, be open subsets of $X$ whose union ...
3
votes
0answers
46 views

Estimation of $\|(1-\Delta)^{\gamma/2}f\|_1$ type

Let $\phi(\xi)\in C_c^\infty(\mathbb{R}^d)$ with value $1$ in a neighborhood of the unit ball and vanishing fast outside the ball. I want to estimate $$\|(1-\Delta)^d (\cdot)^\alpha ...
3
votes
0answers
99 views

Inversion formula for Schwartz-space $\mathcal{S}$.

Let $T\phi = \overline{\mathcal{F}}\mathcal{F}\phi$ for $\phi \in \mathcal{S}(\mathbb{R}^n)$ (rapidly decreasing functions or Schwartz-functions, $\mathcal{F}$ fourier transform and ...
3
votes
0answers
152 views

Solving $ T' = 0 $ for distributions in $\mathbb{R}^n$

Denoting $ T \in \mathcal{D}'(\mathbb{R}^n) $ as distributions with $ T_f(\varphi) = \int_{\mathbb{R}^n} f\varphi\ dx $, I wish to prove the distribution solution of the equation $ T' = 0 $ ...
3
votes
0answers
50 views

Formal integration of a series of the type $-f(x-a)=\sum_{n=0}\frac{C_n}{n!}\delta^{(n)}(x-a)$

This question is inspired from an answer given to this question in the physics stackexchange, specifically the integration step going from (12) to (13). We have a distribution given as ...
3
votes
0answers
335 views

Property of derivative of Dirac delta function in $\mathbb{R}^n$

With reference to Property of Dirac delta function in $\mathbb{R}^n$, is there a similar formula for $\langle g^*\delta', f \rangle$ (or even $\langle g^*\delta^{(n)}, f \rangle$)? By similar I mean a ...
3
votes
0answers
99 views

Colombeau product of distributions

How can use the Colombeau generalized function method to evaluate the product of distributions $ \delta (x) \times \delta (x) $ or $ \delta ^{n} (x) \times \delta ^{m} (x) $ (derivatives of dirac ...
3
votes
0answers
57 views

Why $\partial_{i}(A^{*}u)=A^{*}(\sum^{n}_{j=1}a_{ji}\partial_{j} u)$?

We define the affine transformation on distributions by $$\langle A^{*}u, \phi \rangle=\frac{1}{\det(A)}\langle u,\phi(A^{-1}x)\rangle$$ Assume this we should have $$\langle \partial_{i}(A^{*}u), ...
3
votes
0answers
172 views

Does $\sqrt{\delta^2}$ make more sense than $\delta^2$?

What is the Product of $\delta$ functions with itself? was already asked some time ago. In a comment the OP states: I want to create $\delta(t)$ such that the product with itself is also ...
2
votes
0answers
55 views

Applications of Banach-Alaoglu theorem in the theory of distributions?

Are there some interesting applications of Banach-Alaoglu theorem in the theory of distributions? The theorem provides compact subsets in the $w^*$-topology, so distributions seem a great place for ...
2
votes
0answers
41 views

if $\Delta u = |\nabla u|^{3/2}$ then $u \in C^{\infty}$

Suppose $u \in H^1_{\text{loc}}(\mathbb{R^2})$. I want to show that if $\Delta u = |\nabla u|^{3/2}$ in the distributional sense then $u \in C^{\infty}$. I know that since $\nabla u \in L^2$ (I'll ...
2
votes
0answers
45 views

Fourier transform and non-standard calculus

The Fourier transform is not as easily formalizable as the Fourier series. For example, one needs to introduce tempered distributions to define the Dirac delta-function. Also, it is impossible to view ...
2
votes
0answers
76 views

Besov–Zygmund spaces and the Inverse Function Theorem, is the Inverse Zygmund?

Preliminary Definitions Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
2
votes
0answers
78 views

Nuclearity of $\mathscr{S}$

I have big problems proving that the Schwartz Space $\mathscr{S}(\mathbb{R}^n)$ together with the topology induced by the family $$ \|\varphi\|_{p}:=\sup_{x\in \mathbb{R^n}}\sup_{|\alpha|\leq ...
2
votes
0answers
49 views

Criteria to prove that a map is a tempered distribution

There is any simple sufficient condition to determine if a function is a tempered distribution? For example, given the map : $$ F \phi = \int_\epsilon^\infty \! \frac {\phi(x)}{\sqrt{x}} \, ...
2
votes
0answers
66 views

Is this distributional laplacean a measure?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Suppose that $0<k<t<1$ and $\Phi$ is a mooth function satisfying: $\Phi(x)=0$ for $x\le k$, $\Phi(x)=1$ for $x\ge t$. Take $u\in ...
2
votes
0answers
81 views

Show that the Hilbert transform is a pseudo-differential operator of order 0

I have tried to solve the following exercise but I don't know if I have missed anything. Show that the Hilbert Transform $Hf = \mathcal{F}^{-1}(\operatorname{sgn}(\xi)\hat{f}(\xi))$ is a ...
2
votes
0answers
134 views

Derivative of Dirac delta behavior at 0

I calculated the curl of the vector field $\vec j = \delta(z) f(\rho) \vec e_\phi$ in cylindrical coordinates (ρ,φ,z). For the $\vec e_ρ$ unti vector I got: $$-\delta'(z)f(\rho)$$ The main question ...
2
votes
0answers
144 views

Problem on the integral representation of a tempered distribution

Suppose $\mathscr{S}(\mathbb{R^n})$ is the space of Schwartz functions, in which the seminorms have the form $$\left \| \varphi \right \|_{m}=\underset{\underset{x\in \mathbb{R}^{n}}{|\alpha|\leq m ...
2
votes
0answers
44 views

What is the motivation for “continuity in the sense of distributions”?

Let $M$ be a compact (real) manifold and let $\Omega^m_c(M)$ be the compactly supported $m$-forms on $M$. Apparently a linear map $T : \Omega^m_c(M) \to \mathbb{R}$ is continuous "in the sense of ...
2
votes
0answers
48 views

Multiplying and dividing distributions by non-$C^\infty$ functions.

It's quite easy to see that we can multiply distributions by any $\mathcal C^\infty $ functions. Moreover, if the distribution $T$ is of order $k$, then we can mupliply it by a $\mathcal C^k$ ...
2
votes
0answers
84 views

Fourier analysis on bounded domain?

For tempered distributions on $\mathbb{R}^n$ we can write $\widehat{\nabla f}(p)=p\hat{f}(p)$ and hence by Plancherel, we have equations like $(\nabla f,\nabla g)=(p\hat{f}(p),p\hat{g}(p))$ for ...
2
votes
0answers
304 views

Is there a particular meaning to the sum of Fourier coefficients $a_{n^2}$?

The formula $$\sum_{n=-\infty}^{+\infty} e^{-in^2x}$$ does not converge in any function space but it is perfectly valid in $\mathcal{D}'(\mathbb{R})$. When applied on a test function $\psi(x) = ...
2
votes
0answers
36 views

On the Fourier transform of a certain characteristic function

Consider Schwartz's distribution on $\mathbb{R}^2$. Let $$L=a\partial_x^2+b\partial_x\partial_y+c\partial_y^2$$ and $A:=\{(x,y)\in\mathbb{R}^2|y\geq |x|\}$. The problem asks if $L\chi_A=\delta$ as ...
2
votes
0answers
99 views

What is the set of all functions which can be used as a 'convergence factor' for a Fourier Transform?

At times, I am required to take the Fourier Transform of some function that does not decay quickly enough for the Fourier Transform to converge in the usual sense. For example, $$ ...
2
votes
0answers
118 views

Fourier transform of a tempered distribution

anybody knows how to calculate the fourier transform of $e^{-ax}, a>0$ in the sense of tempered function. I manage to find out it is $ \delta $ (y+ia) but it does not seem right as the 'argument' ...
2
votes
0answers
67 views

Distributions - please check my solution

I have to find a derivative in a distributional sense of the following function (known as Cantor's singular function) $$f(x)=\left\{ \begin{array}{l l l l l} 0, & \quad\text{$ x\leq 0 $}\\ 1, ...
2
votes
0answers
385 views

Show that the Dirac delta distribution can not be represented by a continuous function

How do I show that the Dirac delta distribution cannot be represented by a continuous function? My try is to show that there exists no continuous function $f(x)$ such that $\int f(x) \phi(x) dx = ...