Use this tag for questions about distributions (or generalized 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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12
votes
3answers
1k views

Dirac Delta or Dirac delta function?

Is Dirac delta a function? What is its contribution to analysis? What I know about it: It is infinite at 0 and 0 everywhere else. Its integration is 1 and I know how does it come.
17
votes
2answers
2k views

Sobolev space is an algebra

How do you prove that the Sobolev space $H^s(\mathbb{R}^n)$ is an algebra if $s>\frac{n}{2}$, i.e. if $u,v$ are in $H^s(\mathbb{R}^n)$, then so is $uv$? Actually I think we should also have $||uv||...
7
votes
2answers
181 views

Dirac's delta in 3 dimensions: proof of $\nabla^2(\|\boldsymbol{x}-\boldsymbol{x}_0\|^{-1})=-4\pi\delta(\boldsymbol{x}-\boldsymbol{x}_0)$

If $T_f$ is a distribution, i.e. a linear functional, continuous according to the convergence defined here, defined on the space $K$ of the functions of class $C^\infty$ that are null outside a ...
10
votes
1answer
1k views

Tensor products of functions generate dense subspace?

Let $X$ and $Y$ be two spaces in certain category, $F(\cdot)$ a functor associating each space with a function space (with certain topology). Assume that for any $f\in F(X)$ and $g\in F(Y)$, $f\otimes ...
9
votes
4answers
10k views

what is product of delta function with itself?

what is the product of delta function with itself ? what is the dot product with itself ?
8
votes
6answers
2k views

Distribution theory book

I'm looking for a good book on distribution theory (in the Schwartz sense), I have the basic knowledge as given in Grafakos' Classical Fourier Analysis, but I want to know more about it. Is the ...
3
votes
3answers
417 views

distribution with point support

Let $u$ be a distribution on $\mathbb{R}^n$ with support = $\left\{0\right\}$. Then there exists $N$ such that $u$ has order $N$. Let $\chi\in C_0^{\infty}(\mathbb{R}^n)$ a smooth function with $\chi(...
8
votes
3answers
3k views

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

How does one prove the following identity? $$\int _Vf(\pmb{r})\delta (g(\pmb{r}))d\pmb{r}=\int _S\frac{f(\pmb{r})}{|\text{grad} g(\pmb{r})|}d\sigma$$ where $S$ is the surface inside $V$ where $g(\...
3
votes
1answer
1k views

The constant distribution

If $u$ is a distribution in open set $\Omega\subset \mathbb R^n$ such that ${\partial ^i}u = 0$ for all $i=1,2,\ldots,n$. Then is it necessarily that $u$ is a constant function?
7
votes
4answers
3k views

How to prove that the derivative of Heaviside's unit step function is the Dirac delta?

Here is a problem from Griffith's book Introduction to E&M. Let $\theta(x)$ be the step function $$\theta = \begin{cases} 0, & x \le 0, \\ 1, & x \gt 0. \end{cases} $$ The ...
10
votes
4answers
338 views

Iterated Limits Schizophrenia

Consider the functions $g_n(x)$, with $n\in\mathbb{N}$, $n \ge 1$ and $x\in\mathbb{R}$, defined as follows: $$ g_n(x) = \begin{cases} 2n^2x & \text{if }0 \le x < 1/(2n) \\ ...
1
vote
1answer
517 views

Delta Dirac Function

Show that $\delta$, function of Dirac, defined than $\left<{\delta_0,\phi}\right> = \phi(0)$ belongs to $W^{-1,p}(]-1,1[)$ and $\delta_0 \notin L^p(-1,1)$ $\forall p\geq 1$. How I will be able ...
0
votes
2answers
298 views

Discontinuity of Dirac Delta distribution

I know that the following holds for Step function, but not sure if it holds for the distribution too. Does the following hold for the Dirac delta distribution too? $\delta$ is a linear functional ...
6
votes
2answers
1k views

On distributions over $\mathbb R$ whose derivatives vanishes

Let $I \subset \mathbb R$ be open, $u \in \mathcal D'(I)$ be a distribution whose distributional derivatives vanishes (i.e. is zero for all test functions, which we may assume to be complex valued ...
11
votes
5answers
3k views

Is the Dirac Delta “Function” really a function?

I am given to understand that the Dirac delta function is strictly not a function in the conventional sense and it is a "functional or a distribution". The part which I can not understand why the ...
8
votes
2answers
3k views

Laplacians and Dirac delta functions

It is often quoted in physics textbooks for finding the electric potential using Green's function that $$\nabla ^2 \left(\frac{1}{r}\right)=-4\pi\delta^3({\bf r}),$$ or more generally $$\nabla ^...
5
votes
1answer
530 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 ...
9
votes
1answer
231 views

Growth of Tychonov's Counterexample for Heat Equation Uniqueness

Define a function $\varphi$ on $\mathbb{R}_{+}$ by $$\varphi(t):=\begin{cases}e^{-1/t^{2}}, & {t>0}\\ 0, & {t\leq 0}\end{cases}\tag{1}$$ It is well-known that $\varphi$ is $C^{\infty}(\...
8
votes
4answers
1k views

Dirac delta of nonlinear multivariable arguments

How does one compute a dirac delta function with a multivariable argument? For example, compute: $$ \int^{\infty}_{-\infty}{\rm d}x\,{\rm d}y\, \delta\left(x^{2} + y^{2} - 4\right) \delta\left(\left[...
6
votes
2answers
6k views

Dirac delta in polar coordinates

Given $$x=r\,\cos\theta\\y=r\,\sin\theta$$ and $$x'=r'\,\cos\theta'\\y'=r'\,\sin\theta'$$ how can I express $$\delta(x'-x)\delta(y'-y)$$ in terms of the polar coordinates? And the more general ...
4
votes
1answer
3k views

Clear explanation of heaviside function fourier transform

I know that fourier transform of Heaviside function is : $\hat{H}(x) = \pi \delta(\omega) + i (v.p. \frac{1}{\omega})$ How can i proof this result?
10
votes
0answers
281 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 ...
10
votes
2answers
781 views

Distributions on manifolds

Wikipedia entry on distributions contains a seemingly innocent sentence With minor modifications, one can also define complex-valued distributions, and one can replace $\mathbb{R}^n$ by any (...
3
votes
1answer
1k views

Which functions are tempered distributions?

Today's problem originates in this conversation with Willie Wong about the Fourier transform of a Gaussian function $$g_{\sigma}(x)=e^{-\sigma \lvert x \rvert^2},\quad x \in \mathbb{R}^n;$$ where $\...
3
votes
2answers
654 views

Distribution of infinite order

This is an exercise from Functional Analysis By Walter Rudin on page 178 from chapter Test functions and Distribution. I am having trouble arguing for 2nd part. Question: For $\Omega=(0, \infty)$ $...
3
votes
1answer
282 views

Gauss–Ostrogradsky formula for Distributions

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $u\in W^{1,p}(\Omega)$, $v\in L^{p'}(\Omega)^N$ with $p\in (1,\infty)$. Let $\operatorname{div}(v)$ be the divergence of $v$ in the sense of ...
6
votes
3answers
757 views

Derivatives distribution

Let $f$ be a distribution on $\mathbf{R}^n$ (in the Schwartz sense) such that $$\frac{\partial f}{\partial x_i} = 0 \text{ for $i = 1, \ldots, n$.}$$ Then how to prove that $f$ is a constant? I had ...
5
votes
1answer
161 views

Sobolev spaces in one-dimensional vs multidimensional

Here in Wikipedia, it is said that in the one-dimensional case, it is enough to assume that the $(k-1)$-th derivative of the function $f$, is differentiable almost everywhere and is equal almost ...
5
votes
4answers
5k views

On the derivative of a Heaviside step function being proportional to the Dirac delta function

I am learning Quantum Mechanics, and came across this fact that the derivative of a Heaviside unit step function is Dirac delta function. I understand this intuitively, since the Heaviside unit step ...
4
votes
2answers
734 views

Convergence of test-functions is not induced by any metric.

By $\mathcal{D}(\mathbb{R})$ we denote linear space of smooth compactly supported functions. We say that $\{\varphi_n:n\in\mathbb{N}\}\subset\mathcal{D}(\mathbb{R})$ converges to $\varphi\in\mathcal{D}...
3
votes
1answer
159 views

Fourier transform of unit step function

It is well known that the fourier transform for unit step $U(t)$ is \begin{equation} F(U(t))=\frac{1}{j\omega}+\pi \delta(\omega) \end{equation} When I try to arrive to this expression from the ...
3
votes
1answer
135 views

$\nabla^2(\|\boldsymbol{x}-\boldsymbol{x}_0\|^{-1})=-4\pi\delta(\boldsymbol{x}-\boldsymbol{x}_0)$ with distributions defined on Schwartz space

I know, from a recent enlightening answers received here, that, if we define the distribution represented by Dirac's $\delta$ on the space $K$ of test functions of class $C^\infty$ whose support is ...
3
votes
0answers
146 views

No consistent theory can define a product of distributions: why?

I have been told there cannot be a consistent theory defining a distribution product. Googling for information, I found 1 and 2. Number 1 gives interesting hints on what might happen, and defines a ...
2
votes
1answer
103 views

Products of distributions ill-defined

This question concerns distributions, as often encountered in PDE theory, which are defined as continuous linear functionals on the space $C_0^{\infty}(\Omega)$ of test functions. The product of two ...
0
votes
2answers
97 views

Does delta distribution remain continuous with respect to quasinorm?

I am thinking the accepted answer which is found here: When viewing $\delta: \mathbb{S} \to \mathbb{R}$ (linear and continuous with respect to the usual semi-norms on the Schwartz-space – or ...
-2
votes
1answer
90 views

Is $\delta : \mathcal{S}(\mathbf{R}) \to \mathbf{C}$ continuous with usual seminorm?

I am thinking again the accepted answer which is found here: When viewing $\delta: \mathbf{S} \to \mathbf{R}$ (linear and continuous with respect to the usual semi-norms on the Schwartz-space ...
5
votes
2answers
461 views

Square root of compactly supported C-infinity function

Given $u \in \mathcal{C}^\infty_0(\mathbb{R}^n)$, $u \geq 0$ everywhere, is $v(x) = \sqrt{u(x)}$ also in $\mathcal{C}^\infty_0$? It is clear that the only problematic points are the boundary of the ...
2
votes
1answer
39 views

Given a $C_c^∞(G)$-valued random variable, is $C_c^∞(G)∋φ↦\text E[\langle\xi,φ\rangle]$ an element of the dual space of $C_c^∞(G)$?

Let $G\subseteq\mathbb R^d$ and $$\mathcal D:=C_c^\infty(G)$$ be equipped with some topology $\tau$ $\mathcal D'$ be the dual space of $\mathcal D$ and $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the ...
2
votes
1answer
153 views

Tempered distributions and convolution

I remember that if $f,g \in \mathcal{S}(\mathbb{R}^n)$ , then it is well-defined \begin{align*} \displaystyle (f \ast g)(x)= \int_{\mathbb{R}^n} g(x-y)f(y)dy=\int_{\mathbb{R}^n} (\tau_x \widetilde{g})...
1
vote
0answers
52 views

Represent Dirac distruibution as a combination of derivatives of continuous functions?

It seems from Rudin's functional analysis (P168, Thm6.27), the Dirac distribution $\delta_0$ on $R^1$ can be written as $$ \delta_0=f_0+f'_1+f''_2, $$ for $\{f_i\}_{i=0}^2\in C(R^1)$, there $'$ ...
7
votes
2answers
5k views

Dirac Delta function

I know that $\int_{-\epsilon}^\infty f(x)\delta(x)dx=f(0)$ but what about $\int_0^\infty f(x)\delta(x)dx$? I suppose we have to do this by definition since the lower limit is bang on $0$?
9
votes
1answer
234 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 ...
6
votes
2answers
547 views

Fractional derivatives of delta function $ \delta (x) $

How can I define the fractional derivative of the Delta function? I mean $D^{\alpha}= \frac{d^{\alpha}}{dx^{\alpha}} $ where $\alpha$ can be any real number, then if we define $D^{\alpha} \delta (x) ...
10
votes
2answers
534 views

Origin of the name 'test functions'

This is a very simple question really: where did the name 'test functions', used nowadays when speaking of infinitely differentiable and compactly supported functions, come from? More to the point: is ...
8
votes
1answer
680 views

Sequential and topological duals of test function spaces

Given a test function space, in particular $\mathcal{S}=\mathcal{S}(\mathbb{R}^n)$ (the Schwartz space) or $\mathcal{D}=\mathcal{D}(\mathbb{R}^n)$ (the space of compactly supported smooth test ...
9
votes
2answers
2k views

When can one expect a classical solution of a PDE?

When solving a PDE, there may be a classical solution or a weak solution (or distribution solution). But I am wondering that when people talk about "finding a solution" to some PDE, what do they refer ...
1
vote
1answer
264 views

Dirac delta distribution & integration against locally integrable function

I was reading the a lecture note online about distribution theory and it said: The Dirac delta distribution $\delta \in D'$ is defined as $\delta(\varphi)= \varphi(0) $, and there's no locally ...
8
votes
1answer
136 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
2answers
8k views

Proof of Dirac Delta's sifting property

A common way to characterize the dirac delta function $\delta$ is by the following two properties: $$1)\ \delta(x) = 0\ \ \text{for}\ \ x \neq 0$$ $$2)\ \int_{-\infty}^{\infty}\delta(x)\ dx = 1$$ I ...
5
votes
1answer
170 views

Specific problem on distribution theory.

*****Note: Parts A, C and D I managed. Only need help on part B now would really would appreciate the help on B Hi, in my summer real analysis (or measures and real analysis as my instructor refers ...