Use this tag for questions about Schwartz distributions, also known as Generalised Functions. For questions about "probability distributions", use (probability-distributions). For questions about distributions as sub-bundles of a vector bundle, use (differential-geometry).

learn more… | top users | synonyms

0
votes
1answer
28 views

Non-standard examples of Distributions

Can somebody point me to examples of distributions which are not sums of delta functions or derivatives thereof. Also of course not integrals with local integrable functions. Additional: is there a ...
1
vote
0answers
20 views

Weak convergence in $D^{1,p}(\mathbb{R}^n)$ and in $\mathcal{M}(\mathbb{R}^n)$

What it means: $$u_n\rightharpoonup u ~\text{weakly in }~ D^{1,p}(\mathbb{R}^n)\\ |\nabla(u_n-u)|^p\rightharpoonup\mu ~\text{weakly in}~ \mathcal{M}(\mathbb{R}^n)\\|u_n-u|^{p^{*}}\rightharpoonup\nu ...
2
votes
1answer
26 views

$k^2 e^{ikx} \rightharpoonup 0$ in the Sense of Distributions

So I'm concerned showing $k^2 e^{ikx} \rightharpoonup 0$ in the sense of distributions or in other words for any $\phi \in C_c^{\infty}(\mathbb{R})$ we have for any $\epsilon > 0$ $$ \left\lvert ...
2
votes
1answer
438 views

Fourier transforms of the Heavyside function and the absolute value function

I'm trying to obtain these two Fourier transforms. First of all, I'm using the following definition of Fourier transform: $$\cal{F}(f)(y)=\int_{-\infty}^{\infty}f(x)e^{-ixy}\;dx$$ What I have so ...
-1
votes
1answer
28 views

Kullback-Leibler divergence when the $Q$ distribution has zero values

For discrete probability distributions $P,Q$, the Kullback-Leibler divergence of $Q$ from $P$ is defined to be $$D_{\mathrm{KL}} ( P \mathop{\|} Q ) = \sum_i P(i) \ln \left( \frac{P(i)}{Q(i)} ...
1
vote
1answer
45 views

Differentiating the Dirac Delta distribution

More generally, let $\psi (D)$ denote a pseudodifferential operator on $\mathbb{R}^n$ given by the function $\psi \in S^m_{\rho, \eta}$, the usual symbol class. My question is: can we interpret $\psi ...
0
votes
0answers
10 views

Fourier transformation of Principal value distribution [duplicate]

I have the principal value distribution defined as $pv(\frac{1}{x})(\phi)=\int^\infty_0\frac{1}{x}(\phi(x)-\phi(-x))dx$ and I want to show that the fourier transform is given by $-\pi i\cdot ...
-1
votes
0answers
25 views

Product of delta functions [closed]

What is the result of the following integral? $$\int_{-\infty}^{+\infty}\delta(x-x_1)\delta(x-x_2)f(x)\,dx$$
1
vote
1answer
33 views

Does pointwise convergence imply convergence in distribution? Counterexample?

$f_1,f_2,\dots ,$ and $f$ are in $L_{loc}^1(U)$. I'm trying to give a counterexample where $f_n\to f$ pointwise, but not $f_n\to f$ in $\mathcal{D}^\prime (U)$, where ...
0
votes
2answers
35 views

Does convergence in $L^p$ imply convergence in distribution?

$f_1,f_2,\dots ,$ and $f$ are in $L_{loc}^1(U)$. Prove that if $f_n\in L^p(U)(1\leq p\leq \infty)$ and $f_n\to f$ in the $L^p$ norm or weakly in $L^p$, then $f_n\to f$ in ...
1
vote
0answers
42 views

Negative index Sobolev spaces, properties

I was hoping to find some references for the following facts: Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb{R}^n$. (i) If $\nabla p$ is a distribution in $H^{-1}(\Omega)$ then ...
2
votes
1answer
30 views

Distribution agreeing with function

I'm trying to figure out how to show distributions agree with a given function on some domain. For instance, let $f \in C(\mathbb{R^n}\setminus\{0\})$ such that $f(rx) = r^{-n}f(x)$ and $\int f ...
5
votes
0answers
49 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 ...
2
votes
0answers
50 views

Applications of Banach-Alaoglu theorem in the theory of distributions?

Are there some interesting applications of Banach-Alaoglu theorem in the theory of distributions? The theorem provides compact subsets in the $w^*$-topology, so distributions seem a great place for ...
3
votes
1answer
558 views

Second derivative Dirac delta distribution times $(x-a)^2$, intepretation

I'm not sure if this calculation is correct and if I interpret it correctly (from old exam). Show that $ (x-a)^2 \delta ''_a = 2 \delta _a $. We have for distributions $f$ and test functions ...
0
votes
1answer
33 views

Entire function of exponential type $1$ bounded by $1/(1+|x|)$

Let $f$ be an entire function of exponential type $1$ such that $|f(x)| \leq \frac{1}{1+|x|}$ for all $x\in \mathbb{R}$. First, I have to show that $|f(z)| \leq \frac{Ce^{|Im(z)|}}{1+|z|} z\in ...
1
vote
1answer
34 views

Tempered distributions

Let P be a vector whose componentes are polynomials in $\mathbb{R}^n$ and harmonics. its true that exists a polinomial T that $\nabla T = P$? I think this has something to be with fourie transform, ...
0
votes
0answers
18 views

Expand a distribution to a linear differential form

A linear differential operator $P$ is defined as: $$ PT=\sum_{\alpha=0}^\infty p_{\alpha} \partial^\alpha T $$ A distribution $Q$ which satisfies $P \delta = Q$ is given, where $\delta$ is the Dirac ...
0
votes
1answer
289 views

Delta function with imaginary argument

We have an integral representation for the Dirac delta function as $\delta(x) = \frac{1}{2\pi} \int_{-\infty}^\infty dk e^{ikx}$ . On the other hand, we have for delta function the property: ...
0
votes
1answer
23 views

fourier transform and principal values

Fourier transform and principal values Can anyone tell me from how can i get the fouries transformation of prinicipal value of (1/x) $$p.v\int \frac{1}{x}\Bigg(\int e^{-wix}\varphi(w)dw\Bigg)dx$$
1
vote
1answer
27 views

distributions whose derivative is zero?

I just learned about the notion of tempered distributions $\mathcal{S}'(\mathbb{R})$. But it is unclear that if such a distribution has a 0 derivative (of course in the distribution sense) then it ...
1
vote
1answer
17 views

Distributional solution of this equation

I am having trouble finding the distributional solutions $u$ of: $x^2u = \delta$. Could somebody help? Thanks in advance
0
votes
1answer
19 views

Show that the limit of distributions is Dirac delta

I would like to show that the following statement is true $ \lim_{a\searrow 0} \theta(x)\frac{x^{1-a}}{\Gamma(a)} = \delta(x). $ $\Gamma$ is the gamma function. The above limit is in the sense of ...
0
votes
0answers
22 views

Proving the absolute value of a smooth function is $W^{1,p}$ [duplicate]

How could one prove the following: Take $u \in C^1_c(\mathbb{R}^n)$ Then, $|u|$ is in $W^{1,p}(\mathbb{R}^n)$, $p \in [1;\infty)$. The problem is to show that the derivative in the distribution sens ...
1
vote
1answer
33 views

Convergence of $f_n(x)=n^2f(nx)$ in the sense of distributionas

Let $f$ be a test function such that $\int_{-\infty}^\infty f(x)dx=0$ and $f_n(x)=n^2f(nx)$. Find the distributional limit $\lim_{n\to\infty}f_n$. How can I use the Dominated Convergence Theorem ...
1
vote
1answer
22 views

Distribution equation (x-a)T=0

I have to solve the equation $(x-a)T=0$ , T is a distribution. By definition : $(x-a)\int T(x)\varnothing (x)=0$ I know if I pose $X=x-a$ I find $XT(X)=0$ and $T(X)=\delta(X)$. But I stuck to find ...
3
votes
1answer
31 views

Seminorms in distribution theory are norms, right?

In distribution theory the seminorms that you use there $p_m( \phi) := \max_{|\alpha| \le m} \sup_{x \in \Omega} |(\partial^{\alpha}(\phi) (x)|, \phi \in C_c^{\infty}(\Omega)$. Those guys are norms ...
1
vote
1answer
20 views

Bound on the set of compactly supported distributions with support in the same compact set

Consider the set of all compactly supported distributions $v\in\mathcal{\mathcal{E}^{\prime}}(\mathbb{R}^{n})=\left(C^{\infty}\right)^{*}$ with compact support in a fixed compact set $\Omega$ . ...
4
votes
1answer
94 views

Distributional linear differential equations

What are the most general distributional solutions $u \in \mathcal{D}'(\mathbb{R})$ to $-\frac{d^n u}{dx^n} + c_{n-1}\frac{d^{n-1}u}{dx^{n-1}} + ... + c_0 u = 0$; $-x\frac{d^n u}{dx^n} + ...
1
vote
3answers
2k views

Heaviside step function fourier transform and principal values

I found the following answer on SE: Fourier transform of unit step? However, it is still not clear to me and maybe somebody could explain it clearer. Problem I have the following in my notes of ...
7
votes
3answers
199 views

Distribution theory and differential equations.

How does distribution theory plays role in solving differential equations? This question might seem to be very general. I will try to explain, please bear with me. I understand, distributions make it ...
2
votes
1answer
128 views

Generalized Functions (Distributions) over Manifolds

What is the right way of making sense of generalized functions over manifolds? For concreteness, let me restrict my question to the dirac delta function. The article on Wikipedia on Dirac delta ...
1
vote
1answer
28 views

A variant of the fundamental lemma of calculus of variation

If $F$ is a distribution and its distributional derivative is equal to 0, how can I show that $F$ is (represented by) a constant function i.e. there exists a constant $c$ such that $F(\phi)=c\int\phi$ ...
2
votes
1answer
79 views

$\langle u,\phi\rangle=0$ when ${\rm supp}(u)\cap{\rm supp}(\phi)^\circ=\emptyset$?

I know that $\langle u,\phi\rangle=0$ if ${\rm supp}(u)\cap{\rm supp}(\phi)=\emptyset$. For some while I wondered whether it's enough that $\phi$ vanishes on ${\rm supp}(u)$ but that's not true, as ...
7
votes
0answers
278 views

Connection between distributional and renormalized solutions for Boltzmann equation

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read through some papers by DiPerna and Lions concerning the Cauchy Problem ...
4
votes
5answers
1k 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 ...
2
votes
2answers
174 views

What is wrong in this counter-example?

In reading my textbook, the author give a lemma as follows: Let $X\subset \mathbb{R}^{n}$ be an open set, and let $u\in \mathcal{E}'(X)$ have order $N$. Then $\langle u,\phi \rangle=0$ for all $\phi$ ...
2
votes
1answer
42 views

Sign mistake in Fourier transform of $\frac{x}{1+x^2}$.

I want to calculate the distributional Fourier transform of $u(x) = \frac{x}{1+x^2}$ in one dimension in the distributional sense as $u\notin L^1$. I use the distributional definition of the Fourier ...
1
vote
1answer
50 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: ...
0
votes
0answers
44 views

Complex distributions - what are the appropriate test functions?

In the theoretical physics literature on conformal field theory, one encounters distributional formulas like $$ \frac{1}{\pi}\partial_{\bar z}\frac{1}{z} = \delta(z), $$ where $\partial_{\bar z}$ is ...
5
votes
2answers
41 views

How to prove this simple fact without using distribution theory?

Suppose function $f(x) $ is normalized to unity, i.e., $$ \int dx |f(x)|^2 =1 . $$ Now consider the Fourier transform of $f$, i.e., $$ F(k) = \int d x f(x) e^{-i k x} . $$ Here we assume that $f $ ...
3
votes
3answers
194 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 ...
3
votes
1answer
53 views

Show that the distribution is of the form $C \delta + f$

I'm trying to solve this problem: Let $ u = p.v.(1/x)$, $\phi$, $\psi \in C^{\infty}_c$. I want to show that the distribution $(\phi u )* (\psi u)$ is of the form $C \delta + f$ for some constant C ...
0
votes
0answers
22 views

$\lim\limits_{n\to \infty} \arctan(nx)$ and set-valued limit?

Consider the sequence $a_n(x)=\dfrac{2}{\pi}\arctan(nx)$. $(a_n)$ converges pointwise to $1$ if $x>0$, $-1$ if $x<0$ and $0$ if $x=0$. It does not converge uniformly as the limit function is ...
8
votes
1answer
113 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 ...
1
vote
1answer
69 views

Dirac's delta, infinite series and integral

Why $\int_{-\infty}^x\sum_{i=1}^{+\infty} p^{i-1}\delta(\alpha-i)d\alpha = \sum_{i=1}^{+\infty}\int_{-\infty}^x p^{i-1}\delta(\alpha-i)d\alpha$ where $\delta$ is the Dirac's delta and $p \in ]0;1[$ ...
7
votes
1answer
90 views

Proving that a family of functions limits to the Dirac delta.

For each $\epsilon > 0$, define $f_\epsilon:\mathbb R\to \mathbb R$ as follows: \begin{align} f_\epsilon(k) = \frac{1}{\pi}\frac{\epsilon}{\epsilon^2+k^2}. \end{align} How does one rigorously ...
5
votes
2answers
640 views

Proving the integral of the Dirac delta function is 1

Was wondering if my solution is mathematically accurate enough: The question in the book yields: Derive $$ 1=\int_{-\infty}^{\infty} \delta(x-x_i)\ dx_i $$ From $$ ...
0
votes
1answer
48 views

two dimensional delta function

Is it correct to write $\delta(x,y)=\delta(x)\delta(y)$ where $\delta(x,y)$ is the delta function in two dimensions? Or are there some cases where the above fails to give the correct results when ...
1
vote
0answers
17 views

Fundamental solutions for wave equations which vanish inside characteristic cones

If the number of dimensions $n$ is odd, the flat-space wave operator $-\partial_t^2 + \nabla^2$ admits the fundamental solution $(-\sigma)^{1-n/2} \Theta (-\sigma)$, where $\sigma = - t^2 + |x|^2$. ...