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).

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0answers
8 views

Is there any simple formula for this probability distribution of random walk?

Assume $\{S_n\}_{n\geq 0}$ transits as follows: $S_0=0$, for $k\geq 1$, $P(S_{n+1}=k+1|S_n=k)=\alpha$, $P(S_{n+1}=k|S_n=k)=\beta$ and $P(S_{n+1}=k-1|S_n=k)=1-\alpha-\beta$, where ...
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0answers
15 views

Support of Distribution Function

Suppose I have a distribution function $$C(u,v)$$ with domain $I^2$. Let us define the support of this function as the complement of the union of all open subsets of $I^2$ with C-measure zero. Based ...
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2answers
18 views

Problem on multiplication of distributions with test functions

This question comes from Grubb's Distributions and Operators, question 3.8(b): Consider $u\in \mathcal{D}'(\mathbb{R}^n)$ and $\phi\in C^{\infty}_0(\mathbb{R}^n)$. Find out whether one of the ...
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0answers
21 views

Elementary properties of Fourier transform on the space of tempered distributions.

I'm trying to prove that some of the basic facts of Fourier transform on $L^1(\mathbb{R}^n)$ also holds on the space of tempered distributions. For example: Suppose that $F\in \mathcal{S}^\prime$, ...
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2answers
23 views

Can $\sum_{a = -\infty}^{\infty} e^{i\omega aT_0}$ be represented by dirac delta functions?

The usual definition of dirac delta function says that $\delta(x-\alpha)=\frac{1}{2\pi} \int_{-\infty}^\infty e^{ip(x-\alpha)}\ dp $. The appearance similarity makes me think that it may be possible ...
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1answer
44 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
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0answers
23 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
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1answer
27 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 ...
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
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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
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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
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1answer
43 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
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2answers
40 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 ...
2
votes
1answer
31 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
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0answers
50 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 ...
0
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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 ...
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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 ...
1
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1answer
35 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, ...
2
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0answers
51 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 ...
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1answer
34 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 ...
0
votes
1answer
24 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) $$p.v\int \frac{1}{x}\Bigg(\int e^{-wix}\varphi(w)dw\Bigg)dx$$
2
votes
1answer
28 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
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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
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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 ...
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 ...
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
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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
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1answer
32 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
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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 $\Omega$ . ...
1
vote
1answer
30 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 $F(\phi)=c\int\phi$ ...
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} + ...
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0answers
45 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 ...
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3answers
203 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
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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 ...
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
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 ...
2
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1answer
133 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 ...
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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 ...
2
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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 ...
1
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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
91 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 ...
0
votes
1answer
49 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
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0answers
19 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 $\sigma = - t^2 + |x|^2$. ...
3
votes
0answers
35 views

Distribution induced by a function

We are given $F(x) = |2x+1|, x \in \mathbb{R}$ How to determine whether $$[F|_{(0, \infty)}] \in \mathcal{D}'((0, \infty))$$ $$[F|_{(- \infty, 0)}] \in \mathcal{D}'((- \infty, 0))$$ are regular ...
0
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2answers
26 views

$\text{supp }u =\{0\}$ implies $u=c\delta_0$ in distributional sense?

Given $\text{supp } (u) =\{0\}$ where $u\in D(X)^\prime$ is a distribution and $X\subset \mathbb{R}$. Does this already imply that $u=c \delta_0$? for some constant $c$.
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0answers
20 views

Any way to rewrite/simplify $\int dp ~f(p)~\delta[c-g(p)-E(p)]$?

Im thinking of using $$\delta(g(x)) = \sum_i \frac{\delta(x-x_i)}{|g'(x_i)|}$$ but in my case $$\int dp ~f(p)~\delta[c-g(p)-E(p)]$$ the functions $f$ and $E$ are basically unknown, while $g$ is a ...
3
votes
2answers
58 views

What is the meaning of $1^\lor=\delta$

I tried to look up the definition of a $\lor$ and it does not seem to explain this particular usage $$1^\lor=\delta$$ This is used in a proof that inverse fourier transform of $1$ is $\delta$, but ...
0
votes
1answer
38 views

Inverse of laplacian operator

I recently read a paper, the author treats $$\int_{\mathbb{R}^d}f(y)\cdot \frac{1}{|x-y|^{d-2}}\,dx = (- \Delta)^{-1} f(y)$$ up to a constant in $\mathbb{R}^d$. I am not familiar with unbounded ...
0
votes
1answer
42 views

How do I find the distribution of the laplacian operator acting on Log |f|

Can someone give me some ideas/insight/suggestions on approaching this problem: Calculate the distribution $u(x) = \Delta \log{|\,f\,|}$ where $f$ is a meromorphic function that doesn't vanish ...
0
votes
1answer
64 views

What is ${\cal D}(\Omega_0)\otimes{\cal D}(\Omega_1)$?

How exactly can I prove that ${\cal D}(\Omega_0)\otimes{\cal D}(\Omega_1)$ equals the linear space $M$ spanned by 'simple fucntions' $\phi\in{\cal D}(\Omega_0\times\Omega_1)$ of the form ...