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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3
votes
1answer
70 views

Is every “nice” topological vector space a manifold?

Say $V$ is a topological vector space. What conditions do you need to add on $V$ to make it a (topological, maybe infinite-dimensional) manifold? For instance, can we view the Schwartz class ...
3
votes
1answer
35 views

What does it mean for a distribution to be in $L_2$?

I am new to Sobolev space and distribution theory. So here is what I know. Distributions are linear functionals on $C_0^\infty$. Let's look at the simplest Sobolev space. $H^1(\Omega)$ is equal to the ...
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}} ...
1
vote
0answers
45 views

I need to integrate with $\delta$ against something that isn't a test function!

In relation to ``How does integration over $\delta^{(n)}(x)$ work?,'' I need to evaluate $\int_{-a}^{a}f(x)\delta^{(n)}(x)\, dx$. However, while my $f$ is smooth on its domain, it can't be a test ...
0
votes
1answer
28 views

What test function on a open interval look like?

Let $\phi$ be a function defined on an open interval $I=(a,b)$. The closure of the set of points where $\phi\ne 0$ is called the support of $\phi$. If the support of $\phi$ is a compact set, then ...
3
votes
1answer
38 views

Identification of $L^2$ limits with distributional convergence

I just read the thread on "too much effort" and I would like to be more specific. Is the following reasoning correct: Let $g,g_\delta\in H^1(D)$, $D$ some domain in $\mathbb{R}^n$ with the following ...
3
votes
1answer
64 views

How does integration over $\delta^{(n)}(x)$ work?

For a math paper I need to be able to evaluate $\int_{-a}^{a}\delta^{(n)}(x)\ f(x)\ dx$ for differentiable $f$. I know that it is 'supposed' to equal $(-1)^nf^{(n)}(0)$: $$\int_{-a}^a\delta^{(n)}(x)\ ...
0
votes
0answers
52 views

Infinite solutions of Navier-Stokes equations

Is it a known fact that Navier-Stokes equations have exactly one (possibly infinite) solution in the space of distributions?
0
votes
1answer
121 views

Delta function with imaginary argument

We have an integral representation for the Dirac delta function as $\delta(x) = \frac{1}{2\pi} \int_{-\infty}^\infty dk e^{ikx}$ . On the other hand, we have for delta function the property: ...
1
vote
1answer
35 views

Derivation of a non-continuous function with distribution theory

Consider the following function: $$f(x,t) = \left\{\begin{array}{ll} 1 & x \in [0, t] \\ 0 & x \not\in [0,t] \end{array}\right.$$ What can I say about the derivative of $f$ with respect to ...
0
votes
1answer
81 views

Derivatives of Norms and Absolute Values (distributions)

For example we have for $x \in \mathbb{R}$, $$\frac{\partial}{\partial x}\left| x\right| = 2\Theta(x) -1 $$ and thus $$\frac{\partial^2}{\partial x^2}\left| x\right| = 2\delta(x) $$ We also have, ...
6
votes
2answers
75 views

Is it possible to mathematically explain why solids go under mollification when heated?

Well, I'm sure that many people on MSE might object that this is not a math question, however, I think that there might be a well-posed mathematical answer to this question, or at least I hope so. We ...
3
votes
1answer
159 views

Topologies of test functions and distributions

I'm wondering about some of the topological properties of $\mathcal D(\Omega)$ and $\mathcal D'(\Omega)$: I know $\mathcal D(\Omega)$ is not metrizable, so not first countable (right?). However, my ...
0
votes
0answers
30 views

Fourier transform of a function of characteristic function of a measure

Let $\mu$ be complex measure on $\mathbb{R}^2$ ($|\mu|$ is finite measure) and $\chi$ - its characteristic function $$ \chi(x_1,x_2) = \int_{\mathbb{R}^2} d\mu(p_1,p_2) \exp(i p_1 x_1+i p_2 x_2). $$ ...
2
votes
1answer
34 views

What would be to show $(1+|\xi|^2)^{s/2}\hat{u}\in \mathcal{S}^{'}(\mathbb R^n)$ is a function?

If $s\in\mathbb R$ define the sobolev space $H^s(\mathbb R^n)$ as $$H^s(\mathbb R^n):=\{u\in \mathcal{S}^{'}(\mathbb R^n): (1+|\xi|^2)^{s/2}\hat{u}\in L^2(\mathbb R^n)\}.$$ I have a doubt concearning ...
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 + ...
1
vote
1answer
37 views

Is $L^2(0,T;H^{-1}(\Omega)) \subset \mathcal{D}^*((0,T)\times \Omega)$?

Let $\Omega \subset \mathbb{R}^n$ be a domain. Consider the space of test functions $\mathcal{D}((0,T)\times \Omega)$ and the space of distributions $\mathcal{D}^*((0,T)\times \Omega).$ Is it true ...
3
votes
1answer
109 views

How to build a compact support for a function

I was wondering if it is possible to build a distribution with compact support from a function. More precisely, consider a compact set $\mathbf{K}\subset\mathbb{R}^2\setminus\{0\}$, and a function ...
0
votes
0answers
31 views

Wave front set of a corner

Consider the distribution $u$ defined on $\mathbb{R}^2$ by $$ u(x, y) = \begin{cases} 1 &\text{if } 0 < x < 1;\, 0 < y < 1, \\ 0 &\text{otherwise}.\end{cases} $$ What is the ...
2
votes
1answer
58 views

Weak Laplacian of $\|x\|^\alpha$

Let $\alpha> 0$ and consider the function $\|\mathbf x\|^\alpha = (x^2 + y^2)^{\frac{\alpha}{2}}$ defined on $\mathbb R^2$. I want to compute the Laplacian $\Delta (\|\mathbf x\|^\alpha)$ in the ...
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 ...
2
votes
1answer
69 views

Why the tempered distribution is zero?

My question is derived from the proof of the equation $\Delta f=f$ which has no nonzero solution in $\mathscr{S}'(\mathbb{R}^n)$. The ideal to solve this equation is to use the Fourier transform. By ...
1
vote
1answer
27 views

How to compute the limit of a distribution

I would like to know if the following sequence : $ T_n = \displaystyle \sum_{k=1}^n \dfrac{1}{k^{2}} \delta_{\frac{1}{k}} $ converge in $ \mathcal{D} ' ( \mathbb{R} ) $. If it's converge in $ ...
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 ...
0
votes
0answers
38 views

P.D.E's - Distribution

If we have $$\int_{\Omega} (u_m-u_n) \Delta \varphi \,dx=\int_{\Omega} |u_m-u_n|\,dx$$ for $\Delta\varphi = \operatorname{sgn}(u_m-u_n)$. Now, if I need to choose a test function $\varphi$ such that ...
2
votes
1answer
24 views

Support of the limit of a convergent sequence of distributions

I read that if $u_n \to u \in \mathcal{D}'(X)$, then $$ \text{supp} \, u \subset \bigcap_{n \geq 1} \bigcup_{m \geq n} \text{supp} \, u_m. $$ However, the proof given shows that $$ \text{supp} \, u ...
2
votes
2answers
27 views

Can we conclude that a distribution is a $L^2$ function by testing with $L^2$?

Let $T\colon \mathcal{D}\to\mathbb{R}$ be a distribution. Does $|T(f)|\leq\|f\|_2 \forall f\in\mathcal{D}$ imply $T=T_g$ for some $g\in L^2$? What if $T$ is tempered?
1
vote
2answers
72 views

Derivatives of $|x|$

I wanted to calculate the first and second derivatives of the function $f(x)=|x|$, in order to verify that: $$ f'(x)=\frac{|x|}{x} $$ and $$ f''(x)=2\delta(x). $$ Can you help me?
4
votes
1answer
112 views

Prove that the weak$^*$ topology on the space of tempered distributions is not 1st countable

Please, help me with a proof of this (apparently) known fact whose proof is out of my reach, even though I spent a considerable amount of time looking it up: The weak$^*$ topology on the space of ...
0
votes
1answer
115 views

distribution solution to xT = 0 in Schwartz space

I try to understand the Poisson summation formula from the perspective of distribution theory. However, I got stuck at a problem on the way, namely proving that the distribution solution to $xT = 0$ ...
0
votes
0answers
84 views

Approximate dirac delta and integration error

For a sequence of functions $g_n(x-x_o)$ approximating the Dirac delta I can write: $ \int_a^b g_n(x-x_o) f(x) dx = \int_a^b \delta(x-x_o) f(x) dx + \epsilon_n$ when $x_o \in [a,b]$. I am trying to ...
1
vote
2answers
79 views

Distributional derivatives on hypersurface?

In a paper I was reading, the define a set $Q=(0,T)\times \Omega$, where $\Omega \subset \mathbb{R}^n$ is a bounded domain, and then they write $$\langle \frac{d}{dt}u - \Delta u, \varphi ...
1
vote
0answers
46 views

Convolution of two delta distributions

Show ${\int}_0^{\infty}\delta(x+z)\delta(y-z)dz =\delta(y+x)$ It seems obvious, and I don't think we need a rigorous proof for this (statistical mechanics homework) but I want a rigorous proof of ...
6
votes
1answer
75 views

How to make sense of Fourier series for a distribution?

In particular if I have an array of numbers say, $\{c_m\}_{m\in\mathbb{Z}^n}$. Under what conditions can we say that these are the Fourier coefficients of a distribution? [For examples Bessel's ...
0
votes
2answers
107 views

Delta function multiplied by an exponential function

I do not know if this is an ill-posed question but ... is $\delta(t)e^{-\gamma t}$ equal to $\delta(t)$? Thanks, biologist
3
votes
1answer
66 views

Applications of the theory of distributions outside of PDEs?

Are there any interesting, important or powerful mathematical applications to the Theory of Distributions besides those dealing with partial differential equations?
-2
votes
1answer
48 views

A sequence of functions converging to the Dirac delta

let $g_n(x)=\frac{1}{2}n $ for $|x|<\frac{1}{n}$ and for positive integer n. Prove that $$\lim_{n \to \infty} g_n(x)=\delta(x)$$ Pretty evident after a quick sketch, but I don't know how to show ...
2
votes
1answer
50 views

Weighted Dirac comb as a tempered distribution?

I'm trying to determine when a "weighted" Dirac comb is a tempered distribution. More precisely, trying to prove: $$u=\sum_{k=1}^{\infty}c_k \delta_k\in\mathcal{S}'(\mathbb{R})\iff\exists ...
0
votes
2answers
52 views

A short question concerning the distributional solution of $xf=0$

I was reading my notes on the following result: All the $\mathcal{D}'(\mathbb{R})$ solutions to $xf =0$ are of the form $c\delta $ where $c$ is constant and $\delta$ is the dirac delta distribution ...
1
vote
1answer
64 views

Generalized functions as integral kernels on Hilbert spaces

I'm a physics student and I'm studying functional analysis. I've got a doubt about some operators defined by integral kernels that are generalized functions. Let $L_2(a,b)$ be the Hilbert space of ...
0
votes
1answer
40 views

$(1+x^2)T = 1$ in $\mathscr D'(\mathbb{R})$

I only know that the solutions of $xT=0$ in $D'(\mathbb{R})$ are in the form $c\delta_0$, but I can't figure out how to find general solution to $(1+x^2)T = 1$. Any ideas?
0
votes
0answers
22 views

distribution sense

We have, \begin{eqnarray*} \Psi=\int_0^L \left[\frac{1}{2}\left((1+a^2)u_x^2+2a u_x |u_x|\right)+F(u)+\frac{1}{2}w^2\right]\,dx \end{eqnarray*} We can write, formally, \begin{eqnarray*} \delta ...
0
votes
1answer
53 views

How to prove that limit is equal to zero

How to prove that: $$\lim_{\epsilon\rightarrow 0}(-\log(-x) \phi(x)|_{-\infty}^{-\epsilon} -\log(x) \phi(x)|_{\epsilon}^{+\infty})$$ where $\phi(x) $ is any test function is equal to $0$. It seems ...
2
votes
4answers
159 views

Why is $C_c^\infty(\Omega)$ not a normed space?

I am watching a Coursera video on Théorie des Distributions and I am trying to understand one of the slides. Let $\Omega \subset \mathbb{R}^N$ be an open set and $C_K^\infty(\Omega) = \{ \phi \in ...
2
votes
1answer
53 views

Relationships between growth rates of a distribution and smoothness of its Fourier transform

Let $f\in \mathcal{S}^\prime(\mathbb{R})$ be a tempered distribution, and $\hat{f}$ be its Fourier transform. It is known that when both $f$ and $\hat{f}$ are $L^2$ functions, there are relationships ...
1
vote
1answer
45 views

About a condition for a distribution to be zero

I'm trying to solve the following question: If $S$ is a distribution with compact support on $\mathbb{R}$, which verifies $\langle S, x^n \rangle=0$ $\forall n\in\mathbb{N}$, then $S$ is the ...
3
votes
2answers
95 views

Fundamental solution of a differential operator

I'm trying to solve this question, but I don't know how to deal with it: If we have $b=(b_1,,\dots,b_n)\in\mathbb{R}^n$ and $\beta\in\mathbb{R}$, prove that the differential operator ...
2
votes
1answer
175 views

Fourier transforms of the Heavyside function and the absolute value function

I'm trying to obtain these two Fourier transforms. First of all, I'm using the following definition of Fourier transform: $$\cal{F}(f)(y)=\int_{-\infty}^{\infty}f(x)e^{-ixy}\;dx$$ What I have so ...
1
vote
0answers
37 views

Mellin-Barnes transform of $\frac{1}{\Gamma}$

Let $\varphi \in L_1(c+i\mathbb R)$, where $c > 0$. Then we can define the Mellin-Barnes transform (or the inverse Mellin transform) of the function $\varphi$ by the formula $$ \mathcal ...
1
vote
1answer
60 views

Fourier transform of $\frac{d}{dt}\ln\frac{1}{it}$

I'd like to proove the identity $$\mathcal{F}\left(\frac{d}{dt}\ln\frac{1}{it}\right)=2i\pi H$$ with $H=\mathbb{I}_{\mathbb{R}^+}$ ie the Heaviside step function, $\mathcal{F}$ denote the Fourier ...