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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3
votes
1answer
176 views

Topologies of test functions and distributions

I'm wondering about some of the topological properties of $\mathcal D(\Omega)$ and $\mathcal D'(\Omega)$: I know $\mathcal D(\Omega)$ is not metrizable, so not first countable (right?). However, my ...
0
votes
0answers
32 views

Fourier transform of a function of characteristic function of a measure

Let $\mu$ be complex measure on $\mathbb{R}^2$ ($|\mu|$ is finite measure) and $\chi$ - its characteristic function $$ \chi(x_1,x_2) = \int_{\mathbb{R}^2} d\mu(p_1,p_2) \exp(i p_1 x_1+i p_2 x_2). $$ ...
2
votes
1answer
38 views

What would be to show $(1+|\xi|^2)^{s/2}\hat{u}\in \mathcal{S}^{'}(\mathbb R^n)$ is a function?

If $s\in\mathbb R$ define the sobolev space $H^s(\mathbb R^n)$ as $$H^s(\mathbb R^n):=\{u\in \mathcal{S}^{'}(\mathbb R^n): (1+|\xi|^2)^{s/2}\hat{u}\in L^2(\mathbb R^n)\}.$$ I have a doubt concearning ...
3
votes
0answers
39 views

Exponential of the derivative operator on the Schwartz space?

We consider the derivative operator $\mathrm{D}$ on the space of smooth and rapidly decreasing function $\mathcal{S}$. We denote by $P_n = \frac{1}{0!} + \frac{X}{1!} + \frac{X^2}{2!} + \cdots + ...
1
vote
1answer
40 views

Is $L^2(0,T;H^{-1}(\Omega)) \subset \mathcal{D}^*((0,T)\times \Omega)$?

Let $\Omega \subset \mathbb{R}^n$ be a domain. Consider the space of test functions $\mathcal{D}((0,T)\times \Omega)$ and the space of distributions $\mathcal{D}^*((0,T)\times \Omega).$ Is it true ...
3
votes
1answer
113 views

How to build a compact support for a function

I was wondering if it is possible to build a distribution with compact support from a function. More precisely, consider a compact set $\mathbf{K}\subset\mathbb{R}^2\setminus\{0\}$, and a function ...
0
votes
0answers
32 views

Wave front set of a corner

Consider the distribution $u$ defined on $\mathbb{R}^2$ by $$ u(x, y) = \begin{cases} 1 &\text{if } 0 < x < 1;\, 0 < y < 1, \\ 0 &\text{otherwise}.\end{cases} $$ What is the ...
2
votes
1answer
60 views

Weak Laplacian of $\|x\|^\alpha$

Let $\alpha> 0$ and consider the function $\|\mathbf x\|^\alpha = (x^2 + y^2)^{\frac{\alpha}{2}}$ defined on $\mathbb R^2$. I want to compute the Laplacian $\Delta (\|\mathbf x\|^\alpha)$ in the ...
4
votes
0answers
81 views

Two possible definitions of “vector-valued distribution”

Let $X$ be a reflexive Banach space. Define $$\tag{1} \mathcal{D}^\star(0, T; X)=\left\{ u\colon \mathcal{D}(0, T)\to X\ \text{linear and continuous}\right\} $$ where the topology on the space of ...
2
votes
1answer
76 views

Why the tempered distribution is zero?

My question is derived from the proof of the equation $\Delta f=f$ which has no nonzero solution in $\mathscr{S}'(\mathbb{R}^n)$. The ideal to solve this equation is to use the Fourier transform. By ...
1
vote
1answer
29 views

How to compute the limit of a distribution

I would like to know if the following sequence : $ T_n = \displaystyle \sum_{k=1}^n \dfrac{1}{k^{2}} \delta_{\frac{1}{k}} $ converge in $ \mathcal{D} ' ( \mathbb{R} ) $. If it's converge in $ ...
2
votes
0answers
120 views

Problem on the integral representation of a tempered distribution

Suppose $\mathscr{S}(\mathbb{R^n})$ is the space of Schwartz functions, in which the seminorms have the form $$\left \| \varphi \right \|_{m}=\underset{\underset{x\in \mathbb{R}^{n}}{|\alpha|\leq m ...
0
votes
0answers
38 views

P.D.E's - Distribution

If we have $$\int_{\Omega} (u_m-u_n) \Delta \varphi \,dx=\int_{\Omega} |u_m-u_n|\,dx$$ for $\Delta\varphi = \operatorname{sgn}(u_m-u_n)$. Now, if I need to choose a test function $\varphi$ such that ...
2
votes
1answer
27 views

Support of the limit of a convergent sequence of distributions

I read that if $u_n \to u \in \mathcal{D}'(X)$, then $$ \text{supp} \, u \subset \bigcap_{n \geq 1} \bigcup_{m \geq n} \text{supp} \, u_m. $$ However, the proof given shows that $$ \text{supp} \, u ...
2
votes
2answers
29 views

Can we conclude that a distribution is a $L^2$ function by testing with $L^2$?

Let $T\colon \mathcal{D}\to\mathbb{R}$ be a distribution. Does $|T(f)|\leq\|f\|_2 \forall f\in\mathcal{D}$ imply $T=T_g$ for some $g\in L^2$? What if $T$ is tempered?
1
vote
2answers
75 views

Derivatives of $|x|$

I wanted to calculate the first and second derivatives of the function $f(x)=|x|$, in order to verify that: $$ f'(x)=\frac{|x|}{x} $$ and $$ f''(x)=2\delta(x). $$ Can you help me?
4
votes
1answer
123 views

Prove that the weak$^*$ topology on the space of tempered distributions is not 1st countable

Please, help me with a proof of this (apparently) known fact whose proof is out of my reach, even though I spent a considerable amount of time looking it up: The weak$^*$ topology on the space of ...
0
votes
1answer
125 views

distribution solution to xT = 0 in Schwartz space

I try to understand the Poisson summation formula from the perspective of distribution theory. However, I got stuck at a problem on the way, namely proving that the distribution solution to $xT = 0$ ...
0
votes
0answers
95 views

Approximate dirac delta and integration error

For a sequence of functions $g_n(x-x_o)$ approximating the Dirac delta I can write: $ \int_a^b g_n(x-x_o) f(x) dx = \int_a^b \delta(x-x_o) f(x) dx + \epsilon_n$ when $x_o \in [a,b]$. I am trying to ...
1
vote
2answers
79 views

Distributional derivatives on hypersurface?

In a paper I was reading, the define a set $Q=(0,T)\times \Omega$, where $\Omega \subset \mathbb{R}^n$ is a bounded domain, and then they write $$\langle \frac{d}{dt}u - \Delta u, \varphi ...
1
vote
0answers
48 views

Convolution of two delta distributions

Show ${\int}_0^{\infty}\delta(x+z)\delta(y-z)dz =\delta(y+x)$ It seems obvious, and I don't think we need a rigorous proof for this (statistical mechanics homework) but I want a rigorous proof of ...
6
votes
1answer
94 views

How to make sense of Fourier series for a distribution?

In particular if I have an array of numbers say, $\{c_m\}_{m\in\mathbb{Z}^n}$. Under what conditions can we say that these are the Fourier coefficients of a distribution? [For examples Bessel's ...
0
votes
2answers
124 views

Delta function multiplied by an exponential function

I do not know if this is an ill-posed question but ... is $\delta(t)e^{-\gamma t}$ equal to $\delta(t)$? Thanks, biologist
3
votes
1answer
67 views

Applications of the theory of distributions outside of PDEs?

Are there any interesting, important or powerful mathematical applications to the Theory of Distributions besides those dealing with partial differential equations?
-2
votes
1answer
52 views

A sequence of functions converging to the Dirac delta

let $g_n(x)=\frac{1}{2}n $ for $|x|<\frac{1}{n}$ and for positive integer n. Prove that $$\lim_{n \to \infty} g_n(x)=\delta(x)$$ Pretty evident after a quick sketch, but I don't know how to show ...
2
votes
1answer
55 views

Weighted Dirac comb as a tempered distribution?

I'm trying to determine when a "weighted" Dirac comb is a tempered distribution. More precisely, trying to prove: $$u=\sum_{k=1}^{\infty}c_k \delta_k\in\mathcal{S}'(\mathbb{R})\iff\exists ...
0
votes
2answers
55 views

A short question concerning the distributional solution of $xf=0$

I was reading my notes on the following result: All the $\mathcal{D}'(\mathbb{R})$ solutions to $xf =0$ are of the form $c\delta $ where $c$ is constant and $\delta$ is the dirac delta distribution ...
1
vote
1answer
66 views

Generalized functions as integral kernels on Hilbert spaces

I'm a physics student and I'm studying functional analysis. I've got a doubt about some operators defined by integral kernels that are generalized functions. Let $L_2(a,b)$ be the Hilbert space of ...
0
votes
1answer
41 views

$(1+x^2)T = 1$ in $\mathscr D'(\mathbb{R})$

I only know that the solutions of $xT=0$ in $D'(\mathbb{R})$ are in the form $c\delta_0$, but I can't figure out how to find general solution to $(1+x^2)T = 1$. Any ideas?
0
votes
0answers
23 views

distribution sense

We have, \begin{eqnarray*} \Psi=\int_0^L \left[\frac{1}{2}\left((1+a^2)u_x^2+2a u_x |u_x|\right)+F(u)+\frac{1}{2}w^2\right]\,dx \end{eqnarray*} We can write, formally, \begin{eqnarray*} \delta ...
0
votes
1answer
53 views

How to prove that limit is equal to zero

How to prove that: $$\lim_{\epsilon\rightarrow 0}(-\log(-x) \phi(x)|_{-\infty}^{-\epsilon} -\log(x) \phi(x)|_{\epsilon}^{+\infty})$$ where $\phi(x) $ is any test function is equal to $0$. It seems ...
2
votes
4answers
165 views

Why is $C_c^\infty(\Omega)$ not a normed space?

I am watching a Coursera video on Théorie des Distributions and I am trying to understand one of the slides. Let $\Omega \subset \mathbb{R}^N$ be an open set and $C_K^\infty(\Omega) = \{ \phi \in ...
2
votes
1answer
58 views

Relationships between growth rates of a distribution and smoothness of its Fourier transform

Let $f\in \mathcal{S}^\prime(\mathbb{R})$ be a tempered distribution, and $\hat{f}$ be its Fourier transform. It is known that when both $f$ and $\hat{f}$ are $L^2$ functions, there are relationships ...
1
vote
1answer
47 views

About a condition for a distribution to be zero

I'm trying to solve the following question: If $S$ is a distribution with compact support on $\mathbb{R}$, which verifies $\langle S, x^n \rangle=0$ $\forall n\in\mathbb{N}$, then $S$ is the ...
3
votes
2answers
101 views

Fundamental solution of a differential operator

I'm trying to solve this question, but I don't know how to deal with it: If we have $b=(b_1,,\dots,b_n)\in\mathbb{R}^n$ and $\beta\in\mathbb{R}$, prove that the differential operator ...
2
votes
1answer
236 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
vote
0answers
38 views

Mellin-Barnes transform of $\frac{1}{\Gamma}$

Let $\varphi \in L_1(c+i\mathbb R)$, where $c > 0$. Then we can define the Mellin-Barnes transform (or the inverse Mellin transform) of the function $\varphi$ by the formula $$ \mathcal ...
1
vote
1answer
63 views

Fourier transform of $\frac{d}{dt}\ln\frac{1}{it}$

I'd like to proove the identity $$\mathcal{F}\left(\frac{d}{dt}\ln\frac{1}{it}\right)=2i\pi H$$ with $H=\mathbb{I}_{\mathbb{R}^+}$ ie the Heaviside step function, $\mathcal{F}$ denote the Fourier ...
1
vote
0answers
70 views

Does the norm exist on $\mathscr S' \subset \mathscr D'$?

I take an extension from $L^2(\mathbb R)$ to $\mathscr S'(\mathbb R)$ of tempered distributions for a mapping of nonlinear distribution. I do not want to use seminorm, but the norm, therefore the ...
2
votes
2answers
59 views

$xT' = 1$ in $D'(\mathbb{R})$

I need help solving the following problem: I want to show that all solutions of $$xT' = 1\ , T \in D'(\mathbb{R})$$ take the following form: $c_{1} + c_{2}1_{[0, \infty)} + ln|.|$ What I tried so far ...
0
votes
1answer
118 views

Extend from the spaces $L^2$ to the tempered distributions $\mathscr S'$

Recall that $\mathscr S(\mathbb R) \subset L^2(\mathbb R)$. Assume the continous and bounded functions $f: \mathbb R \to \mathbb C, \,f \in C(\mathbb R), f(t) \not= 0$ only if $a \leq t \leq b$ and ...
1
vote
2answers
108 views

Convolution and Dirac delta [closed]

I need to prove the following: $\delta_0 * \phi = \phi$, where $\phi$ is a test function. Thank you for your help.
2
votes
0answers
38 views

What is the motivation for “continuity in the sense of distributions”?

Let $M$ be a compact (real) manifold and let $\Omega^m_c(M)$ be the compactly supported $m$-forms on $M$. Apparently a linear map $T : \Omega^m_c(M) \to \mathbb{R}$ is continuous "in the sense of ...
1
vote
1answer
63 views

How to solve distributional equation?

What are the only solutions of a distributional equation: $$xT'=0$$ Thanks. Any hint? I know that $T'(\phi)=-T(\Phi')$.
2
votes
1answer
39 views

Translation is continuous

Let $\mathcal D$ be the space of 'test-functions'. Those are infinitely differentiable functions with compact support. Define the following convergence on $\mathcal D$. $(\phi_j) \to \phi$ in ...
0
votes
0answers
19 views

What is the modulus of smoothness for Wigner-Ville Distribution?

I heard today a seminar where the speaker talked about General Shannon Sampling operators and its modulus of smoothness. I can only find this article about the modulus of smoothness. I think you can ...
0
votes
0answers
18 views

Boundary of real part of functions in $H^p$ and Poisson nontangential maximal function

I have two questions when reading on $H^p$ spaces, many books do not give their proofs. First we reminde that $H^p(\mathbb R^2_+)$ consists of all functions $F$ which is analytic in the upper half ...
0
votes
2answers
161 views

Fourier transform of a unity function and of unit step function

Fourier transform of the unity function is the Dirac delta distribution. I think this means: In particular, the Fourier transform of the unity function is the Dirac delta distribution, ...
-1
votes
1answer
61 views

Is delta distribution continuous and differentiable with dual space norm?

I know that delta distribution $\delta : \mathcal S (\mathbf R) \to \mathbf C$ is continuous with usual seminorm and here. I am interested in its continuity with dual-space $H^{-1}(\Omega)$ of ...
1
vote
0answers
46 views

What is the difference between these two kernel definitions?

I am reading my graft and the document of David Haussler about Convolution Kernels on Discrete Structures, UCSC-CRL-99-10. My graft and the other document The terminology seems to differ. The ...