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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11
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0answers
86 views

Normal form of currents

(question now crossposted to mathoverflow ) Let $M$ be an $n$-dimensional manifold. Then the space of currents $\mathcal D^k(M)$ of degree $k$ on $M$ is the space of continuous linear functionals on ...
0
votes
1answer
21 views

Is $F(x)= \frac{1}{|x|^{r}}, (x\in \mathbb R)$ a distributiuon?

Does it make sense to talk of $F(x)= \frac{1}{|x|^{r}}, (x\in \mathbb R)$ for some $r>0$ in the sense of distribution? (I am just confused about the origin) (I mean, Is $F$ a distribution? If yes, ...
3
votes
2answers
62 views

Fourier Transform leading to $\delta$: How does the Integration work?

So it is well-known that the complex exponential $$f(t) = e^{i\omega_0t}$$ has Fourier transform $$F(\omega) = 2\pi \delta(\omega-\omega_0) \ .$$ The transformation integral $$F(\omega) = ...
2
votes
0answers
24 views

The equivalent of “tempered distributions” for the Mellin transform?

The Fourier transform is defined for tempered distributions. For these distributions, the test functions are those functions decreasing more quickly at $\pm \infty$ than $|x|^{-n}$ for all n. In ...
0
votes
1answer
38 views

Distributional derivative of $2$-variable indicator function

Let us say we have the indicator function $\chi_{\{|x|\leq 1\}}$ in $\mathbb{R}^2$. How can I write out the weak derivative of this indicator function? Is it $\delta_{|x|=1}$? Or it should be vector ...
3
votes
0answers
43 views

How do I show that a distribution is a function?

While reading Grigor'yan's book on the heat kernel I have encountered the following definition of a Sobolev space on a Riemannian manifold $M$: $W^2 (M) = \{ u \in W^1 (M) : \Delta u \in L^2 (M) ...
2
votes
1answer
60 views

Does this distribution exist?

Consider the following summation: $$\sum_{n,d} \frac{δ\left(t - \tfrac{n}{d}\right)}{(nd)^σ}$$ Where $δ(t)$ is the Dirac delta distribution, and $σ$ is a constant, and $n,d>0$. Every partial sum ...
1
vote
0answers
21 views

Understanding the embedding of $W^{\infty, 2}$

I am trying to understand Sobolev's embedding theorem, more precisely to understand when a Sobolev generalized function of infinite order is smooth of some order. Consider the following statement ...
1
vote
0answers
29 views

problem 9.26 from Folland's real analysis Fourier Transform of $ G(x,t) = (4t\pi)^{(-n/2)} e^{{-|x|^2}/{4t}} \chi_{(0,\infty)}(t)$

I was just given this question from Folland's real analysis second edition dealing with tempered distributions and their Fourier transforms Exercise 26 on page 300 : On $ R^n \times R $ let $ ...
6
votes
0answers
78 views

Intuition and slick proof that distributions are special cases of hyperfunctions?

I am looking for some slick or simple proof that distributions are a special case of hyperfunctions. Furthermore, what is the intuition behind this fact? Why should one think of hyperfunctions as ...
3
votes
2answers
61 views

Mistake in reasoning about Sobolev spaces

I am new to Sobolev spaces and, while trying to construct a proof, I make some subtle mistake that I cannot detect. The setting: let $C \subset \Bbb R^n$ be a closed, measure-$0$ set. Let $U = \Bbb R ...
1
vote
1answer
33 views

Finding inverse Laplace transform of a fraction of polynomials

I am trying to find the inverse Laplace transform of $$\frac{4s^3 + s}{s^2+1}$$ I tried polynomial long division and reduced it to the following expression: $$4s - \frac{3s}{s^2+1}$$ But I'm not ...
1
vote
1answer
25 views

How do divergences of vector fields generate distributions?

Just to clarify the title before I start, there are some "fuzzy" words that I want to get out of the way: Divergence here is in the sense of the divergence theorem, the operator sometimes written ...
1
vote
1answer
62 views

Gauss' law in differential form for a point charge

I'm trying to understand how the integral form is derived from the differential form of Gauss' law. I have several issues: 1) The law states that $ \nabla\cdot E=\frac{1}{\epsilon 0}\rho$, but when ...
2
votes
0answers
70 views

Cauchy principal value for solving the integral of complex exponential

I need to solve the following integral (if it is possible): $$\int_0^{\infty}dx\,f(x) \left\{ \lim_{t \rightarrow \infty}\int_0^{t}e^{i(x-x_0) \tau}d\, \tau \right \}$$ I found an expression in an ...
2
votes
1answer
41 views

Differential equation for distributions(where did I go wrong?)

I was trying to solve the following problem(where "$u\in D'(I)$" means that $u$ is a distribution on the interval $I$): Find all solutions $u\in D'(\mathbb{R})$ to the following equation: ...
1
vote
1answer
32 views

Question regarding elementary distribution theory

Let $D'(I)$ be the space of distributions on an open interval $I$, and let $D(I)$ be the space of test functions on $I$. I got the following homework assignment: "Define $u\in D'(\mathbb{R})$ be ...
2
votes
1answer
27 views

Mellin transform of rescaled delta distributions

There's something about the Mellin transform I don't get, so hopefully someone can tell me what it is that I'm doing wrong. Let's define the Mellin transform of $f(t)$ as $\mathcal{M}\{f(t)\}(s) = ...
2
votes
3answers
54 views

Can distributions be thought of as functions of a real variable?

I understand that, given some function space, distributions lie in the dual space. In that sense, they can be thought of as functions of a "function of a real variable" variable. But the common ...
2
votes
1answer
45 views

Hankel transform of a Bessel function of different order

Here I found that $$ \int_0^\infty J_\nu(kr) J_\nu(sr) r dr = \frac{\delta(k - s)}{s} = \frac{1}{s^2}\delta\left(1 - \frac{k}{s}\right). $$ I wonder how can that be derived and if a similar method can ...
0
votes
0answers
42 views

Can we make $S_n \to \delta_x$ for $S_n$ an exponential polynomial?

Consider $f_\lambda: \Bbb{R}_+ \to \Bbb{R}_+$,$$f_\lambda(t) = e^{-\lambda t}$$ Now consider the finite linear combinations of these functions (exponential polynomials) $$ S(t) = \sum_{i = 1}^N ...
5
votes
0answers
186 views

Laplace transform of functions related to type $\mathcal{S}$, and the relation to entire functions

I have doubts in the following two questions : What is the Laplace transform of $[x^k\varphi(x)]^{(q)}$, where $\varphi\in \mathcal{S}_\alpha^\beta$ and $-\infty<x<\infty$ , ...
0
votes
1answer
60 views

Integral (Fourier transform) of Heaviside radial function in 3D

I am trying to calculate the following integral: $ \int \frac{d k_x d k_y d k_z}{(2 \pi)^3} \left[ \exp( - \frac{(k_x^2 + k_y^2 + k_z^2) \sigma^2}{2}) + \frac{1}{2} H(\sqrt{k_x^2 + k_y^2 + k_z^2} - ...
4
votes
0answers
45 views

“Contradiction” to Bochner's theorem for distributions

I recently asked a question "For what values of $\lambda$ the distribution $(x-i\epsilon)^{\lambda}$ is positive?". User Marcel was kind enough to point out in his answer that one uses Bochner's ...
1
vote
1answer
134 views

Reference for “distributional derivative being zero implies being constant”

I know that if a distribution (generalized function) has zero derivative, then it is a constant. I also know the proof. But I have a hard time finding a reference which contains a statement of this ...
2
votes
0answers
53 views

Another equivalent characterization of Schwartz function?

Suppose I have a smooth function $\psi$ from $\mathbb{R}^n$ to $\mathbb{C}$, for which I know that $$ \sup_{x\in\mathbb{R}^n}\left||x|^k\Delta^{p}\psi(x)\right|<\infty $$ for all ...
1
vote
1answer
42 views

The Fourier transform of a tempered distribution is supported at the origin

If the Fourier transform of a tempered distribution $G$ is supported at the origin, does this imply that $G$ is a constant? Can anyone give a reference or a short argument?
1
vote
0answers
80 views

Extending an identity for the Dirac delta function

The identity $$x^p \; \delta^{(n)}(x) = (-1)^p \frac{n!}{(n-p)!} \; \delta^{(n-p)}(x)$$ can easily be derived from the generalized Leibnitz formula for $n$ and $p$ positive integers: $$\int \; x^p ...
1
vote
0answers
52 views

Distributions and non-smooth functions

Is it possible to define distributions acting on non-smooth functions? The reason I'm asking is because of the rendering equation $$L_o(x,\omega_o) = \int_\Omega f_r(x,\omega_o,\omega_i) ...
2
votes
0answers
31 views

Deriving a certain delta-sequence with respect to its index

At the end of some calculations I've reached $$\lim \limits _{t \to 0_+} \int \limits _{\Bbb R ^n} \frac {h(t,x,y)} t f(y) \Bbb d y$$ where $$h(t,x,y) = \frac {\Bbb e ^{\frac {\Bbb i |x-y|^2} ...
2
votes
2answers
62 views

How to take derivatives of a convolution when the kernel's derivative is in the distribution sense?

I came need to take the derivative of the following convolution: $$ \int_{-\infty}^\infty \operatorname{sgn}(x-y)e^{-|x-y|}f(y) \, dy $$ However, the derivative of the kernel only exists in the sense ...
1
vote
0answers
62 views

How to show that $\int_{-\infty}^{\infty} \mathrm{d}^3 \textbf{k} \frac {e^{i \textbf{k x}}} {(2 \pi)^3} = \delta^3(x)$ in spherical coordinates?

Recently I had to deal with Fourier transformations and delta functions, and I was wondering how about that. I know, that its trivial to show in cartesian coordinates, but i couldn't do it in ...
2
votes
1answer
72 views

Why is the support of Dirac distribution $\{0\}$?

Distributions are of two types: those that are obtained from locally integrable functions, and those that aren't. For the first type, the support of distribution is simply the support of the function. ...
0
votes
1answer
36 views

Way to think about weak deriviate

Something hit me when I read the definiton of weak derivite. Would it be right to think about the weak deriviate in terms of distributons, i.e that the distribution $\int f \phi$ induced by f in $ ...
2
votes
1answer
56 views

Problem in distribution theory and tempered distributions

I just encountered this question in my real analysis class involving distribution theory it is question 25 chapter 9 from Folland's real analysis second edition, which reads as follows: Suppose ...
4
votes
1answer
82 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 ...
0
votes
0answers
57 views

Dirac Delta Distribution and non-compactly supported test function

I would like to know if there is any problem with defining the following expression: $$ I = \int_0^\infty g(t) \delta(f(t))\mathrm{d}t $$ where $0<\lim\limits_{t\to\infty} g(t) =L<\infty$ and ...
8
votes
0answers
133 views

What plays the role of the identity for the generalized convolution associated to the Fourier-Bessel transform?

In traditional Fourier theory, the Dirac delta plays the role of an "identity" for the $L^1$ algebra with respect to the usual convolution. The convolution is traditionally built out of group ...
3
votes
1answer
102 views

Lebesgue integral of Dirac delta

If I recall correctly, for a bounded function $f$ $$ \int_{\mathbb{R}} f \, d\mu = \int_{\mathbb{R} \setminus \{ a \} } f \, d\mu + f(a) \mu (a).$$ For the Lebesgue measure, $\mu(a) = 0$ and $$ ...
2
votes
0answers
65 views

What are some properties of the sheaf of distributions?

In a course on measure theory, the lecturer proved that distributions (on a locally convex space I think) form a sheaf $\mathcal D$. He isn't interested in sheaves, so he didn't elaborate. Afterwards, ...
0
votes
1answer
50 views

different generalized functions?

I am trying to solve a PDE that's order 1 in time $t\ge0$ and order 2 in space $x\ge0$. The solution $u(x,t)$ exists, is unique and possesses the following properties: $u(x,t)\ge0$ for all ...
2
votes
0answers
52 views

Schwartz impossibility result

I was wondering what made it impossible to define a product of distributions. Googling, I found two questions, one of which stated the following impossibility result: There is no associative ...
3
votes
0answers
132 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 ...
3
votes
1answer
55 views

Weak convergence and integrals

Assume $$u_k\rightharpoonup u,\quad v_k\rightharpoonup v\quad\text{in}\quad L^1(0,T;Y)\tag{1}$$ and $$\int_0^T u_k(t)\varphi'(t)\ dt=-\int_0^T v_k(t)\varphi(t)\ dt\tag{2}$$ for some $\varphi\in ...
5
votes
1answer
94 views

Fourier transform of the Heaviside function

As you can see from the title I want to calculate the Fourier transform of the Heaviside function $u(t)$. Proven the the Heaviside function is a tempered distribution I must evaluate: $$ \langle ...
0
votes
1answer
120 views

Is this operator a Fourier multiplier operator?

I want to study the Fourier transform of $$L_{\alpha}(t) = \frac{e^{i\alpha t}}{t^2} - i\frac{\alpha}{t}$$ Basically i am trying to get a grip on, given a $f$, what is $f(t)\ast L_{\alpha}(t)$ and am ...
3
votes
3answers
80 views

Definition in Lax “sequence of continuous functions tending to $\delta$”, are distributions needed for understanding?

I'm trying to read Lax's functional analysis. In chapter 11 he makes a definition which I don't like. A sequence of continuous functions ${k_n}$ on a $[-1,1]$ tends to $\delta$ if $\int_{-1}^{1} ...
0
votes
1answer
85 views

How to make a change of variable inside the Dirac delta?

Let $\delta(\phi) = \phi(0)$ be the dirac delta. I would like to compute $\int_{\mathbb{R}} h(x) \delta(\lambda x) dx$ 1) Since $\delta$ is an unit mass on $0$ $$\int_{\mathbb{R}} h(x) ...
8
votes
1answer
200 views

Do Electrical Engineering Researchers Usually Know Higher Math? e.g. Measure and Distribution Theory, Functional Analysis

I am an EE undergraduate student, and I recently took two courses in signals and systems. We were exposed to things like the Dirac delta (without mention of distributions), Fourier series (without ...
3
votes
1answer
56 views

Convolution of tempered distributions where one has compact support.

For $u\in\mathcal E'(\mathbb R^n)$ and $v\in\mathcal S'(\mathbb R^n)$, we defined $u\ast v$ by $\langle u\ast v, \phi\rangle = \langle v, \check u \ast \phi \rangle$ for all $\phi\in\mathcal S(\mathbb ...