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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0
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1answer
64 views

Limit of $t^2 \cos t x$ as distribution.

I am interested in finding the limit $t^2 \cos t x$ as $t \rightarrow \infty$ in the sense of schwartz distributions. After some integration by parts I get $$( t^2 \cos t x,\phi)=\int \cos(x)\phi(x/...
2
votes
0answers
95 views

Show that the Hilbert transform is a pseudo-differential operator of order 0

I have tried to solve the following exercise but I don't know if I have missed anything. Show that the Hilbert Transform $Hf = \mathcal{F}^{-1}(\operatorname{sgn}(\xi)\hat{f}(\xi))$ is a pseudo-...
1
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0answers
63 views

Extension of a Pseudodifferential Operator

Let $M$ be a smooth manifold with countable atlas, and define the distributions $\mathscr{D}'(M)$ as the dual space to the smooth densities with compact support, and $\mathcal{E}'(M)$ as the dual ...
1
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1answer
158 views

Find distribution solving a differential equation

I think I have solved the following differential equation, but I am not sure of all steps are justified. Exercise: Find all distributions $u \in \mathcal{D}'(\mathbb{R})$ such that $x(u' -u) = \...
1
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0answers
61 views

Finding weak solutions

I am trying to understand how to solve differential equations of distributions. For example if one consider $ u' + u = \delta_{0}$, where $ u \in \mathcal{D}'$, this would correspond to $<u, -\...
2
votes
1answer
43 views

Duals of embeddings in the space of distributions

If $ \Lambda \colon X \hookrightarrow \mathcal{D}'$ is a continuous embedding of a normed vector space $X$ into the space of distributions (for example $X=L^p$), is it true that the dual of $\Lambda(X)...
0
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1answer
80 views

Find limit of a sequence of distributions

I am trying to solve the following exercise: Determine the limit in $\mathcal{D}'(\mathbb{R})$ of $\lim_{t\rightarrow \infty} t^{2}xe^{itx}$, $u_{t} = t^{2}xe^{itx}$. I have tried evaluating the ...
-2
votes
1answer
117 views

eigenvalues to Laplacian

Let $\Omega$ be a bounded subset of $\mathbb R^d$ and let $g\in L^2(\Omega )$. Let $(\lambda_n)_{n\in \mathbb{N}}$ be the eigenvalues of the Laplacian operator $\Delta$, and $(e_n)_n$ the ...
3
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0answers
52 views

Do we have $C^\infty \cap \mathcal{O}_C' = \mathcal{S}$ and/or $C^\infty \cap \mathcal{S}' = \mathcal{O}_M$?

We define the following traditional function spaces from distribution theory. $\mathcal{S}$ the space of rapidly decreasing smooth functions. $\mathcal{S}'$ the space of tempered distributions, dual ...
2
votes
1answer
91 views

Poisson summation formula for the Casimir effect

I'm studying the Casimir Effect at finite temperature. To calculate the Helmoltz free energy in the canonical ensemble I need to sum a particular series. In some scientific papers it is suggested to ...
0
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0answers
59 views

Question with tried Eigenvalues of Laplacian operator and Sobolev spaces III.

Let $\Omega$ be a bounded subset of $\mathbb R^d$ and let $g\in L^2(\Omega )$. Let $(\lambda_n)_{n\in \mathbb{N}}$ be the eigenvalues of the Laplacian operator $\Delta$, and $(e_n)_n$ the ...
2
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1answer
58 views

Why do test functions have to be real valued?

Seems like if functions can be made to be complex-valued, they're going to be complex-valued. I have not seen anything about why test functions must be real valued.
1
vote
2answers
254 views

Eignvalues of Laplacian operator and Sobolev spaces

Let $\Omega$ an open bounded in $\mathbb{R}^n$ and $I$ un interval in $\mathbb{R}$, let $t_0 \in I$, $f\in H^1_0(\Omega)$, $g\in L^2(\Omega)$. Let $(\lambda_n)_{n\in \mathbb{N}}$ the eigen values od ...
0
votes
0answers
85 views

Is $2\delta(x) \neq \delta(x)$?

$2\delta(x) \neq \delta(x)$ since, by definition, Can this been seen graphically though? If so, how? If not, why is it that they are mathematically different but graphically the same? Btw, I ...
1
vote
1answer
76 views

show a PDE has no distribution solution in $\mathbb{R^2}$\{0}

" Consider the following equation in the plane $x\frac{\partial u}{\partial y}-y\frac{\partial u}{\partial x}=f(x^2+y^2)$ where $f(t)$ is a $C^\infty$ function of the real variable $t$ such that $...
1
vote
1answer
70 views

Fourier transform of $\frac{x_i^2}{|x|^2}$

For a function $f$ in $L^1(\mathbb{R}^n)$, it is natural to define the Fourier transform as $$\mathscr{F}(f)(\xi)=\int_{\mathbb{R}^n}f(x)e^{-ix\cdot \xi}dx.$$ And the we may extend it to rapidly ...
3
votes
0answers
217 views

Does the fundamental theorem of calculus hold for BV functions?

I am a bit confused and I hope you can help me in understanding a bit better these things. Let us start by considering one dimensional case. Let $f\colon \mathbb (a,b) \to \mathbb R$ be a function. ...
0
votes
1answer
112 views

Verify that a function is a solution to the 3-dimensional wave equation.

For $n=3$. How to verify that a smooth function $$ u(.,t)=f*\partial_t\Phi_t+g*\Phi_t \tag{*} $$ is a solution to the 3-dimensional wave equation $$ u_{tt}=\Delta{u} \quad u(x,0)=f(x) \quad u_t(x,0)=g(...
0
votes
0answers
155 views

Dilation and translation of the Dirac Delta distribution

Given $\delta(ax) = \frac{1}{|a|}\delta(x)$ and $\delta(ax-b) = \frac{1}{|a|}\delta(x-b/a)$ , it it true also that $\delta(a(x-b)) = \frac{1}{|a|}\delta(x-b)$ ?
1
vote
1answer
273 views

The issue of treating an inverse Fourier transform in terms of a tempered distribution.

Consider the wave equation $$ u_{tt}=\Delta{u} \quad u(x,0)=f(x) \quad u_t(x,0)=g(x) \tag{*} $$ A solution to this equation is given by $$ u(.,t)=f*\partial_t\Phi_t+g*\Phi_t \tag{**} $$ where $\Phi_t$...
1
vote
1answer
126 views

convolution -questions

I'm lost, can you help me please. How compute the product convolution between two distributions $T$ and $S$? (we suppose that $T * S$ exist)? How we compute the product convolution between an ...
2
votes
1answer
49 views

Convolution computing

How we can compute the convolution product $$\Big(\sum_{n=0}^{+\infty} \delta_n^{(n)}\Big) \star \Big(\sum_{n=0}^{+\infty} \delta_n\Big)$$ where $\delta$ is Dirac distribution? Thank's for the help
1
vote
1answer
84 views

Computing Fourier transform of a surface measure.

I am almost beginner in the topics of Fourier transform. So, I am asking this question here. Let $n=3$ and let $\mu_t$ denote surface measure on the sphere $|x|=t$. Then how do we show that $$ \frac{\...
0
votes
1answer
158 views

Understanding a use of Leibniz Formula, functional analysis

I am working through the following text: http://homepage.ntlworld.com/ivan.wilde/notes/gf/gf.pdf Several times the author cites 'Leibniz Formula' which I understand as the product rule. I am not sure ...
1
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1answer
70 views

equation in Sobolev space

i have this exercice: Let $f\in L^2(\mathbb{R}^n)$. 1- Prouve that the equation $\Delta u - u = \dfrac{\partial f}{\partial x_i}$ admeit a unique solution $u \in H^1(\mathbb{R}^n)$? 2- Prouve the ...
1
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0answers
53 views

Smoothness of distributions

I've reached an impasse in reading some texts on distribution theory, as several of them mention smooth distributions, but none of them actually define what it means. Therefore I'd like to know if ...
1
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1answer
47 views

Examples of functions with values in distributions

What is an example of a function in $L^p((0,T);\mathcal{D}'(\mathcal{R}))$? I ask this because the Majda-Bertozzi book on Incompressible flow deals with vortex sheet initial data $\omega(t,\cdot)\in \...
2
votes
0answers
164 views

Derivative of Dirac delta behavior at 0

I calculated the curl of the vector field $\vec j = \delta(z) f(\rho) \vec e_\phi$ in cylindrical coordinates (ρ,φ,z). For the $\vec e_ρ$ unti vector I got: $$-\delta'(z)f(\rho)$$ The main question ...
1
vote
1answer
67 views

Holomorphic Schwartz-space-valued function

I was trying the last few days to generalize the Paley-Wiener theorem in a quite obvious direction... or so I thought. The original Paley-Wiener theorem talks about functions and distributions with ...
0
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0answers
48 views

Compactly supported distribution is finite linear combination of …

How to show that every compactly supported distribution can be expressed as a finite linear combination of (distributional) derivatives of functions in ${C^k_0({\bf R})}$, for any fixed $k$?
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0answers
80 views

Method of PDE solution by Fourier transform

In Rudin's Functional Analysis (Chapter 7, exercise 17), Rudin claims that for $n=1$ or $2$, if $u$ is a distribution on $R^n$ with compact support $K$, whose Fourier transform $\hat{u}$ is a bounded ...
1
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1answer
72 views

A distribution of order k+1 is sum of two.

Let $\lambda$ be a compactly supported distribution of order $k+1$ then how to prove $\lambda=\delta'+\mu$ for some compactly supported distributions of order $k$?
2
votes
1answer
70 views

Calderón-Zygmund $\times$ Schwartz $=$ Calderón-Zygmund

I am in a functional analysis class, and we are being asked to show that if $\eta$ is a Schwartz function and $K$ is a Calderón-Zygmund distribution, then their product is also a Calderón-Zygmund ...
3
votes
1answer
74 views

The $\frac{1}{x+i\varepsilon}$ distribution.

I read that the distribution defined as: $$ \lim_{\varepsilon \rightarrow 0}\frac{1}{x+i\varepsilon}$$ is equal to $$p.v. \frac{1}{x} -i\pi \delta(x)$$ So that for any test function $f$, $$\lim_{\...
0
votes
1answer
22 views

Close fourier transforms implies close time domain functions?

What conditions do we need so that $A(f)\approx B(f) \Rightarrow \mathcal{F}^{-1}\left\{A\right\}(t)\approx\mathcal{F}^{-1}\left\{B\right\}(t)$ The Fourier transforms of my two things look alike. In ...
0
votes
1answer
44 views

$f, \hat {f} \in L^{p} \cap L^{\infty} \implies f\in A(\mathbb R)$?

$1\leq p \leq \infty $. We put, $$X_{p}= \{f\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R) :\hat{f}\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R)\};$$ and we consider the algebra of Fourier ...
1
vote
2answers
189 views

Strange Dirac delta distribution contradiction

Consider the following integrals in variables $x,y$ over the whole $\mathbb{R}$, where $a,b\in\mathbb{R}/0$ are constants: $$\int dx \int dy ~\delta(x-a)\delta(y-b\,x)=\int dy ~\delta(y-b\,a)=1$$ In ...
1
vote
1answer
48 views

Expected distance within a distribution is smaller?

consider we have two general distributions $f_1$ and $f_2$, assume they have different support $S_1$ and $S_2$. Is the expected distance btween two points draw from the same distribution smaller than ...
1
vote
1answer
72 views

How to prove a tempered distribution always has a order?

I just learn some distribution theory. In Friedlander's book, "Introduction to the theory of distributions" (page 97), he said: The dual of $\mathscr{S}(R^n)$ (Schwartz Space) consists of ...
0
votes
1answer
104 views

Generalized functions and vector calculus theorems

To apply the divergence theorem, for example, there are conditions on your function. Your function must be a function in the ordinary sense in the first place. But in Electrodynamics, sometimes our ...
3
votes
0answers
267 views

Weak convergence implies uniform convergence in distribution?

Consider an empirical processes $\{v_T(\theta), T\geq 1 \}$ and assume it is weakly convergent to the stochastic process $\{ v(\theta); \theta \in \Theta\}$., i.e. $$ E^*(f(v_T(\theta)))\rightarrow E(...
2
votes
1answer
103 views

Understanding Tempered Distributions

I'm reading about tempered distributions in Folland's Real Analysis. In particular, tempered distributions are defined as the class of linear functionals on a Schwarz space. On page 293, Folland says: ...
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0answers
50 views

Variable density in the equation of motion

At a fixed point in time, consider the equation of motion $$ \nabla \cdot \boldsymbol \sigma(u) + \boldsymbol f = \rho \ddot{\boldsymbol u} \quad \text{in $\Omega \subset \mathbb R^d$} $$ for a ...
1
vote
1answer
56 views

When is the composition of a function with Dirac delta a valid distribution?

If $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is nicely behaved, one can view $\delta(f)$ as a distribution (linear functional on $C^{\infty}_c(\mathbb{R}^k)$)- but what if you don't have nicely behaved ...
2
votes
1answer
67 views

Recover Fundamental solution of wave equation on $\mathbb{R}^n$ by on the sphere

It's well known that $\frac{\sin{t\sqrt{-\Delta}}}{\sqrt{-\Delta}}\delta$, the fundamental solution of wave equation on the $\mathbb{R}^n$ can be expressed as the form \begin{equation} \lim_{t\to 0}...
1
vote
1answer
171 views

Relationship between convergence in the space of bump functions $\mathcal{D}(\mathbb{R}^n)$ and the Schwartz space $\mathcal{S}(\mathbb{R})^n$

Given the space of bump functions $\mathcal{D}(\mathbb{R}^n)$, smooth and with compact support, $$\mathcal{D}(\mathbb{R}^n) := \{ v \in C^\infty(\mathbb{R}^n) : \mathrm{supp}(v) \ \mathrm{is\ compact}...
3
votes
1answer
61 views

Limit of a functional

I'd like to find: $$ \lim_{\varepsilon\rightarrow 0}\frac{\varepsilon}{\varepsilon^2+x^2}\qquad \mbox{ in }\mathcal D'(\mathbb{R}) $$ And I started with the definition: $$ \left\langle \frac{\...
2
votes
1answer
59 views

the continuous functions with norm

I'm having trouble trying to understand what does means the first expression in particular the last term in it should we add $\|f\|_{\infty} \leq \infty$ or what i can't see what is his role ($\|f\|...
5
votes
2answers
146 views

Uniqueness for tempered distributional Cauchy problems

Question. Assume that $U\in C^1(\,[0, \infty)\to \mathcal{S}'(\mathbb{R}^n)\,)$ is a solution to the following tempered distributional Cauchy problem $$\tag{CP}\begin{cases} \frac{ d U}{dt} = f \...
3
votes
2answers
235 views

Why Dirac's Delta is not an ordinary function?

Given the following definition of the Dirac's Delta: $$\delta: \mathcal{D}(\mathbb{R}^n) \to \mathbb{R}: \varphi \mapsto \langle \delta,\varphi \rangle = \varphi(\mathbf{0})$$ where $\mathcal{D}(\...