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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2
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1answer
82 views

Distributional derivative of absolute value function

I'm tying to understand distributional derivatives. That's why I'm trying to calculate the distributional derivative of $|x|$, but I got a little confused. I know that a weak derivative would be ...
0
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1answer
23 views

boundedness of $\mathcal{S}(\mathbb{R})$ functions

I was told that every function in the schwartz space is bounded, i.e. If $f\in \mathcal{S}(\mathbb{R})$, then one can show $f$ is bounded by some $C$. Could anyone show me the explicit calculation? ...
1
vote
1answer
43 views

dirac delta function limit form equality

Show that $$\lim_{y\to\infty}\frac{1}{\pi}\frac{y}{y^2+x^2} = \delta(x)$$ I do not know where the $\pi$ arise.
0
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1answer
43 views

Series of tempered distributions converge to dirac distribution

I want to show that for a tempered distribution $u$ and a series $\psi_k$ of smooth functions such that $\psi_k(x)=1$ for $\vert x\vert\leq 2^{-k}$ and $\psi_k=0$ for $\vert x\vert\geq2^{-k+1}$ there ...
0
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1answer
32 views

an inequality in the proof of properties of mollifier does not hold for $p=\infty$

This is part of the proof of mollifier properties in Evans PDE, which has been posted numerous times... In the above, the inequality does not hold for $p=\infty$, why? (in the case of ...
1
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1answer
41 views

Evans PDE p.714 Change of variable and change of integration region

In the following definition of convolution involving mollification . When I make change of variable $x-y=z$, I have ...
1
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1answer
20 views

2-dimensional delta function (complex plane)

I have a task to show that $$\partial_{\bar{z}} \frac{1}{z - \zeta} = \pi \delta^{(2)}(z - \zeta) $$ But I thought, that delta-function is determined by $\int f(\zeta) \delta(z-\zeta) d\zeta = ...
0
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0answers
34 views

Laplace transform and distributions

I was studying for my course in Fourier Analysis and was going through some old exams, when this question came up: Let $s^{-1}_+$ and $s^{-1}_-$ denote the analytic distributions given by the ...
1
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0answers
34 views

distribution space

What are the differences between function spaces and distribution spaces? I was reading enter link description here But I do not know what are these distribution spaces and how they differ from ...
0
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1answer
37 views

Homogeneous Littlewood-Paley decomposition

I have a question concerning Littlewood-paley-theory. Suppose we have test functions $\psi_k$ supported in annuli $\{2^{k-1}\leq\vert\xi\vert\leq2^{k+1}\}$ such that ...
1
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1answer
16 views

Smoothness of integrals of dirac delta function

I found this text online: "In general, integrating the $\delta$ function or one of its integrals makes it smoother. Differentiating it increases the discontinuities. For example $\int\delta $ is ...
9
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0answers
163 views

Half Solved: A problem on the heat operator not being elliptic with a weakened version of elliptic regularity

I should first mention this: in my studies of Sobolev spaces I have completed all the questions of chapter 9 from Folland's real analysis with the help of this site and this is my last one, which is ...
1
vote
1answer
31 views

Operator satisfying $\langle Pu, \phi\rangle=\langle u, ^tP\phi\rangle$?

Let $\Omega\subseteq \mathbb R^n$ be an open subset and $P:\mathscr{D}^\prime(\Omega)\longrightarrow \mathscr{D}^\prime(\Omega)$ be a continuous linear operator. Is it true that there exists only one ...
3
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0answers
39 views

Is $\sum_n\exp(ian+ibn^2+icn^3)$ known in terms of anything else?

For arbitrary $a,b,c$, does the series $$F(a,b,c)=\sum_{n=-\infty}^\infty\exp\left(ian+ibn^2+icn^3\right),$$ i.e. an evenly-weighed series of exponentials of cubic polynomials, converge to anything ...
0
votes
2answers
60 views

Integral with $\delta$ function

I got this integral in a quantum field theory problem: $$ \int\limits_{-\infty}^{+\infty}\!\!\! dp \, \frac{p^2 \delta\left(\sqrt{p^2-m_2^2}+\sqrt{p^2-m_3^2} -m_1\right)}{\sqrt{p^2-m_2^2} ...
0
votes
0answers
27 views

What is the result of Heaviside function times the Dirac delta function?

As the title suggest, I am looking to understand which is the result of such operation. In fact, I am willing to find the root (numerically) of: $$ \sqrt{(f(x)H(f(x)))^2+(g(x)H(g(x)))^2}-K=0 $$ The ...
4
votes
1answer
51 views

Advantange of having a complete topology on test functions

Let's consider $\mathscr D(\Omega)$, the space of test functions on $\Omega \neq \emptyset \subseteq \mathbb R^n$ as usually defined. For the sake of clareness, $$\mathscr D(\Omega) = \cup_K ...
4
votes
1answer
47 views

Taking the divergence of a field with a singularity when $\vec{r}=0$ produces a Dirac's delta.

I'm currently taking a classical electrodynamics course. I have a mathematical background and I know that the classical theorems of integral calculus (Stokes, Gauss, ...) are just particular versions ...
1
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1answer
37 views

The containment between the Schwartz space, its dual, and the Lebesgue space $L^2$

I read from my note that $$\mathcal{S}(\mathbb{R})\subset L^2(\mathbb{R})\subset\mathcal{S}'(\mathbb{R}).$$ Where $\mathcal{S}$ is the space of rapidly decreasing function on $\mathbb{R}$, ...
0
votes
1answer
59 views

Identity for rescaled Dirac Delta, $\delta(kx)$

I´m trying to proof the following Statement. $$\delta(kx)=\frac{1}{|k|} \delta(x).$$ I already tried to proof and I got this. $$u=kx \Rightarrow x=\frac{u}{k},dx=\frac{1}{k} du \\ ...
2
votes
1answer
84 views

Some intuition on a specific problem on Sobolev's embedding theorem with its relation to Fourier transform of restricted functions

I have recently encountered this problem in my studies of Sobolev spaces and generalized functions (distributions), on which I can say I might have some intuition but cannot stumble across a final ...
0
votes
2answers
35 views

How to evaluate products involving the delta function and Cauchy principal value?

Prove that $x\delta(x) = 0$ and $xP(\frac{1}{x})=1$ Here $P$ means the Cauchy principal value. How can I start this? And if I prove the second, will $xP(\frac{1}{|x|})=1$ also follow? ...
2
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0answers
44 views

Integration by parts involving a special definition of a Dirac delta-distribution

given a "definition" of a $\delta$-function as follows $\int dz \, f(x,z) \, f^{-1} (y,z) = \delta (x-y)$ , I would like to know how to apply knowledge over this to solve an integral like $\int dz ...
0
votes
1answer
42 views

Derivative of absolute distribuion

$(T',\phi) = -(T,\phi')$ is the definition of derivative of distribution function $T$ How to use this to evaluate: $e^{|x|}$ $\sin|x|$ P($\frac{1}{x}$) In 3, it is the cauchy principal value. Can ...
0
votes
1answer
25 views

Differentiable everywhere but at a point where it has a discontinuity with a jump proof

$g(x)$ is a differentiable function everywhere but at a point $x_{0}$ where it has a discontinuity with a jump: $\Delta g_{0} = \lim_{\epsilon \to 0} [g(x_{0}+ \epsilon)-g(x_{0}- \epsilon)]$ How to ...
0
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0answers
62 views

dirac delta, cauchy principal value and step function

P - cauchy principal value and is: $P\left(\frac{1}{|x|},f\right) = \int_{|x|<1}\frac{f(x)-f(0)}{|x|}dx + \int_{|x|>1}\frac{f(x)}{|x|}dx$ Show that the solution to $v(x) = c\delta(x) + ...
1
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0answers
23 views

Criterium forSubspace of tempered distributions

I have a question concerning the subspace $\mathcal{S}'_h$ of tempered distributions defined by $u\in\mathcal{S}'_h\Leftrightarrow\lim_{\lambda\rightarrow\infty}\Vert\theta(\lambda ...
0
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0answers
43 views

Identity of dirac delta function

Show that if f is continuous $\frac{d}{dx}(f(x)\delta(x)) = f(0)\delta'(x)$ if f is differentiable, use Leibnitz rule to conclude that $\frac{d}{dx}(f(x)\delta(x)) = ...
1
vote
1answer
61 views

A question involving sharpening the bound on Sobolev type inequality with Sobolev spaces in terms of distributions of Schwartz functions

I have met this problem recently in my real analysis class involving sharpening the bound on a Sobolev type inequality, from Folland's real analysis, but first I should mention the notations used ...
0
votes
2answers
75 views

Integrating the product of the Heaviside function with an exponential

This is a question from my textbook in an applied mathematics class: On $\mathbb{R}$ with fixed $\alpha >0$, consider the sequences of nonegative continuous functions $$f_k(x) = k^\alpha ...
0
votes
1answer
38 views

Prove there do not exists such distribution.

If $u\in \mathcal{D}'(0, \infty)$ such that $$\langle u,\varphi\rangle=\sum^{\infty}_{n=1}\varphi^{(n)}\Big(\frac{1}{n}\Big)$$ Prove that there do not exists any $v\in \mathcal{D}'(\mathbb{R})$ whose ...
0
votes
1answer
29 views

Proof that the Laplacian of Poisson's fundamental solution is zero everywhere except at the origin?

I know that these things are better calculated using spherical coordinates and/or Fourier transforms. I tried to calculate this in cartesian coordinates anyway: Let $G(\mathbf{x},\mathbf{x_0}) = ...
7
votes
1answer
133 views

Only two parts left : Problem on Fourier Transform and convergence of Tempered Distributions

I recently met this problem from Folland's real analysis second edition involving a specific question on distributions (exercise 19 page 299) which reads as follows: On $ R $ let $ F_0 = ...
1
vote
1answer
36 views

Problem on convergence in distributions from Folland's real analysis

I just met this problem from Folland's real analysis involving the theory of distributions (generalized functions) and their Fourier transform, exercise 15 page 291 which reads: Define G on $ R^n ...
4
votes
3answers
158 views

Why is $f(x) \delta(x) = f(0)\delta(x)$ only true when $x=0$?

This is a follow up from a previous question asked by me. I know that $$\delta(x) = \begin{cases} 0 & \space \mathrm{for} \space x \ne 0 \\\infty&\ \mathrm{for} \space x = 0 \end{cases} $$ ...
1
vote
1answer
25 views

the solution to heat equation in convolution form

Let $g\in C(\mathbb{R})\cap L^{\infty}(\mathbb{R})$. Let $u$ be defined as the function $$u(t,x)=\int_{\mathbb{R}}p_t(x-y)g(y)\,dy$$, where $$p_t(x)=\frac{1}{\sqrt{4\pi t}}e^{-\frac{|x|^2}{4t}},\quad ...
1
vote
0answers
115 views

wave front set - directions of singularities

I am learning about the wave front set of a distribution but am having difficulty understanding some details, which to me seem counter intuitive. We know the fourier transform of a smooth function ...
1
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0answers
44 views

Fourier transforms of distributions

I am reading a proof claiming that every partial differential operator $P(D)$ has a fundamental solution $E$. It says that "if we have a distribution $u$ on $R^n$ with $u(P(D)\phi)=\phi(0)$ for ...
9
votes
2answers
166 views

Is $\int |x\rangle \langle x|dx$ Mathematical?

I am enrolling in a Quantum Mechanics class. As we all know, the formulation of the basic ideas from QM relies heavily on the notion of Hilbert Space. I decide to take the course since it might help ...
3
votes
0answers
21 views

Equivalent descriptions of Sobolev spaces on compact manifolds

While reading through a set of lectures on the Laplacian on manifolds, I encountered two descriptions of Sobolev spaces. The first one, valid only for compact manifolds (because it needs to globalize ...
0
votes
1answer
47 views

Math major transferred to Electrical Engineering, trying to bridge the gap [closed]

I did a minor in mathematics a couple years ago and the non-engineering (i.e. rigorous) math I have been exposed to were two proper courses in prob and statistics, 2 courses in real analysis and 2 ...
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0answers
16 views

Equation $f=Pu+a\cdot u$ in $\mathscr{D}^\prime(\mathbb T^n)$ and fundamental solutions?

Suppose I have the following equation $$f=Pu+a\cdot u \quad \quad (1)$$ in $\mathscr{D}^\prime(\mathbb T^n)$ where $f$ is a distribution induced by a smooth function $f\in C^\infty(\mathbb T^n)$ and ...
4
votes
1answer
97 views

The Green's function of the beam deflection equation

This is a problem in a textbook used in my class: Suppose we have an infinite elastic beam, where the deflection $u(x)$ satisfies the differential equation $$\frac{d^4 u}{dx^4}+k^4 u = > ...
1
vote
1answer
53 views

Rudin's application of the mean value theorem

I am studying theorem 6.26 (page 152) in Rudin's "Functional Analysis" that presents distributions as derivatives of continuous functions. Right at the beginning of the proof, if $\Omega$ is the ...
1
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0answers
16 views

Is the map $\mathbb R^n\longrightarrow \mathscr{D}^\prime(\mathbb R^n)$, $x\longmapsto k(x, \cdot)$, continuous?

Recall, $a\in C^\infty(\mathbb R^n\times \mathbb R^n)$ is a symbol of order $m\in\mathbb R$ if for every $\alpha, \beta\in\mathbb N_0^n$ there is $C_{\alpha, \beta}>0$ such that ...
4
votes
0answers
49 views

Computing the Fourier transform of the distribution $\|x\|^{-\alpha}$.

Question: Suppose we are given the tempered distribution $\|x\|^{-\alpha}$. We want to compute the Fourier transform $\mathcal{F}[\|x\|^{-\alpha}](\xi)$. What techniques are available for ...
11
votes
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 ...