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).

learn more… | top users | synonyms

8
votes
0answers
80 views

Delta distributions with nonlinear arguments

I am confused by the use of nonlinear arguments with the Dirac $\delta$ distribution that I am encountering in the literature. This looks like a widespread use, but for concreteness let us focus on a ...
7
votes
0answers
94 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 ...
6
votes
0answers
243 views

Idea behind distributional solutions

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read thorugh some papers by DiPerna and Lions concerning the Cauchy Problem ...
5
votes
0answers
166 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
83 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
63 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
73 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
131 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
36 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
41 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
35 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
67 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 ...
3
votes
0answers
60 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 ...
3
votes
0answers
89 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
45 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
123 views

Computing the Fourier transform (in the sense of tempered distributions) from a sequence of samples

Let $f\in L_{\operatorname{loc}}^2 (\mathbb{R})$ be a function of period $2\pi$. I'd like to get some help proving the following identity: $$\hat{f}=\sum_{n=-\infty}^{\infty} \hat{f}(n)\delta_n$$ ...
3
votes
0answers
131 views

Delta function and integrating over level sets?

Consider the three-dimensional integral $$ \int_{\mathbb R^3} d^3x\,f(x)\delta(g(x)) $$ where $\delta$ is the dirac delta, $f,b:\mathbb R^3\to\mathbb R$ and $g(x) = 0$ on some surface $S$. Is there ...
3
votes
0answers
47 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
320 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
92 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
54 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
165 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
57 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
22 views

Poisson summation formula for the Casimir effect

I'm studying the Casimir Effect at finite temperature. To calculate the Helmoltz free energy in the canonical ensemble I need to sum a particular series. In some scientific papers it is suggested to ...
2
votes
0answers
101 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
44 views

A catch with Dirac Delta Function

We know that $$ \int_{\mathbb{R}} f(t)\delta(t) \mathrm{d}t = f(0) $$ if $f$ is continuous. What will it be if $f$ is not continuous? For instance, what is the value of $$ \int_{\mathbb{R}} ...
2
votes
0answers
110 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
34 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
36 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
57 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
78 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} ...
2
votes
0answers
198 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
30 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
88 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
97 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
93 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 ...
2
votes
0answers
64 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
272 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 = ...
2
votes
0answers
139 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 $ ...
2
votes
0answers
91 views

Dirac $\delta\left( \left[\sqrt{p^2+m^2}-\sqrt{k^2+p^2+2\cdot k\cdot p\cos(\theta)}\right]^2 -k^2-m^2 \right)$

This question is related to $f(k) = 0$, but we now we consider $\delta(f(k))$, i.e. $\delta\left( \left[\sqrt{p^2+m^2}-\sqrt{k^2+p^2+2\cdot k\cdot p\cos(\theta)}\right]^2 -k^2-m^2 \right)$ We ...
2
votes
0answers
95 views

Distributions over locally compact Abelian groups: when can they be Fourier transformed?

Pontryagin duality shows us every locally compact Abelian group---such as $\mathbb{R}^n$, $\mathbb{Z}$, the circle $\mathbb{R}/\mathbb{Z}$ or any finite Abelian group---has a Fourier transform. In the ...
2
votes
0answers
68 views

Does multiplication commute with taking of fundamental solution (heat equation)

Let $\Phi(t,x)$ be a heat function, $$ \Phi(t,x) = \frac{1}{\sqrt{4 \pi t}} \exp\left(-\frac{x^2}{4t}\right). $$ Then $(\partial_{t} - \partial_{xx})\Phi(t,x) = \delta(t)\delta(x)$. Furthermore, ...
2
votes
0answers
51 views

Representation of distribution by nonnegative measure

Let $T \in \mathcal{D}'(\mathbb{R}_{+})$ be a distribution on $\mathbb{R}_{+}$ such that for any $f \in \mathcal{D}(\mathbb{R}_{+})$, $f \geqslant 0$ we have $$ \langle f, T \rangle \geqslant 0 $$ ...
2
votes
0answers
79 views

Weak derivative and homeomorphisms commute

Suppose $\phi:H^1(S) \to H^2(R)$ is a linear homeomorphism, where $S$ and $R$ are compact surfaces in $\mathbb{R}^n$. Let $\varphi \in D(0,T;H^1(S))$ be a $H^1(S)$-valued $C_c^\infty(0,T)$ function. ...
2
votes
0answers
605 views

Distributions supported on a single point

Let $d=1$. (i) Show that if $\lambda$ is a distribution and $n\geq1$ is an integer, then $\lambda x^n=0$ if and only if $\lambda$ is a linear combination of $\delta:=\delta_{\{0\}}$ and its first ...
2
votes
0answers
96 views

Liouville's Theorem in $\mathbb{R}^n$

Liouville's Theorem states that if a tempered distribution is harmonic, $\Delta{u}=0$, then $u$ is given by a polynomial. For the argument, we take Fourier transform of $\Delta{u}=0$ to obtain ...
2
votes
0answers
94 views

Why does the following define a distribution and of which order?

I want to show that $$\phi\mapsto\underset{\varepsilon\searrow 0}{lim}\int_{-\infty}^{\infty}\frac{\phi(x)}{x+i\varepsilon}dx$$ defines a distribution on $\mathcal{D}(\mathbb{R})$ but I just don't ...
2
votes
0answers
54 views

Action of linear functional on integral depending of parameter

Let $K(x,\omega) \in C^{\infty}(\Omega \times \Omega)$, where $\Omega$ is a domain in $\mathbb{R}^{n}$. Let $\mu$ be a probability measure on $\Omega$. My question is under which conditions an ...
2
votes
0answers
206 views

Representation of compactly supported distribution

Is this true? Any compactly supported distribution $T\in \cal D'$ can be represented as finite sum of partial derivatives of functions.
2
votes
0answers
241 views

The topology of distributions

I have been wondering about the following concerning the spaces $\mathcal D$ of test functions (say on $\mathbf R$). It is my understanding that the topology on this space is inductive limit ...