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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1answer
83 views

Uniform convergence in the proof of properties of mollifier (Evan's approach)

I am still trying to understand Evans' proof on the properties of mollifier. In the proof of (iii), I understand that the crux of the proof is that uniform continuous function on compact set is ...
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1answer
51 views

Schwartz functions have finite $L^p$ norm

It is known that the Schwartz space is dense in $L^p$. And I was told that Schwartz functions are bounded in $L^p$. Could anyone show me "Every Schwartz function is bounded in $L^p$" by explicitly ...
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0answers
20 views

Sum of normed function spaces equipped with two-parameter family of norms, Dual inequality

Let $(\cdot,\cdot)$ be the $L^2(\Omega)$ scalar product, and let $V=L^1(\Omega)$, $W=H^{-1}(\Omega)$ (the dual space of $H_0^1(\Omega)$). My question is if there exists a constant $C$, such that for ...
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1answer
203 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? ...
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1answer
49 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.
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1answer
57 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 ...
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1answer
36 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 ...
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1answer
47 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 ...
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1answer
26 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 = ...
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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 ...
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0answers
38 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 ...
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1answer
44 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 ...
2
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1answer
22 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 ...
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0answers
178 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
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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 ...
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0answers
44 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 ...
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2answers
62 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} ...
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0answers
44 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
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1answer
66 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
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1answer
75 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 ...
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1answer
41 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 $\mathbb{R}$, ...
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1answer
68 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 \\ ...
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1answer
88 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 ...
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2answers
37 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? ...
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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
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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 ...
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1answer
26 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 ...
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0answers
78 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) + ...
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0answers
24 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 ...
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0answers
50 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)) = ...
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1answer
65 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 ...
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2answers
125 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
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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
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1answer
34 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
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1answer
144 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 = ...
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1answer
55 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 ...
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3answers
181 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
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1answer
27 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
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0answers
122 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 ...
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0answers
45 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 ...
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2answers
178 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 ...
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0answers
31 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
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1answer
56 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
17 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
149 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
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1answer
65 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 ...
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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
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0answers
57 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 ...
13
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1answer
104 views

Normal form of currents

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 the space of test $n-k$-forms. Two typical ...