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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2
votes
1answer
134 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
42 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
189 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 }}...
2
votes
1answer
42 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
37 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
89 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
328 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
369 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
165 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
100 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 \rangle_{\...
1
vote
0answers
120 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 ...
7
votes
1answer
330 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
674 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
103 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?
0
votes
1answer
750 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
124 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 N\in\mathbb{...
0
votes
2answers
73 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
104 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
45 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
1answer
68 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
364 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
99 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
55 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
211 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 $b_1\frac{\...
3
votes
1answer
703 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
1answer
85 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 M_c^{-1}...
1
vote
1answer
70 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
111 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
74 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
135 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 ...
2
votes
2answers
411 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
55 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
105 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
60 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
2answers
490 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, $\mathcal ...
-1
votes
1answer
93 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 $H_{0}^...
1
vote
0answers
62 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 ...
2
votes
1answer
199 views

What is the interpretation of $\delta(x)\ln\delta(x)$, where $\delta(x)$ is the Dirac's delta function?

What's the result of the following integral? $$\int f(x)\delta(x)\ln\delta(x)\mathrm{d}x$$ where $f(x)$ is a smooth function (continuous derivatives of as high order as needed).
1
vote
1answer
82 views

When is $W^{m,p}(\Omega)$ dense in $L^p(\Omega)$?

Let $\Omega\subset\mathbb{R}$, $m\in\mathbb{N}$ and $1\leq p<\infty$. What are the (most common) sufficient conditions (if such conditions exists) that we can impose on $\Omega$, $m$ and $p$ to ...
1
vote
1answer
47 views

Distribution of a product of RVs

I have this question which I cannot figure out where I was doing it wrong. Let $(X,Y)$ be a jointly continuous RV with density function $$f_{(X,Y)}(x,y)=\frac{12.5}{LW}\;\;\text{for}\; \;0.9L\leq x\...
1
vote
1answer
179 views

Fundamental solution of nonlinear PDE

A fundamental solution of a linear PDE (in sense of Schwartz), $Lu=0$ is defined as a distribution $E$ such that $LE=\delta$. Now I wish to find fundamental solution of nonlinear PDE, such as the ...
2
votes
2answers
114 views

Computation of integral involving Heaviside function

Let $H : \mathbb{R} \to \mathbb{R}$ denote the Heaviside function: $$ H(y) = \begin{cases} 0 & y < 0, \\ 1 & y \ge 0. \end{cases} $$ Suppose that $c > 1$ is fixed with $t$ ...
0
votes
1answer
46 views

Quadratic Time-Frequency Representation with L2 norm

I have been reading the book Multidimensional Particle Swarm Optimization for Machine Learning and Pattern Recognition. They use different norm for different problems in their automatic ECG detection ...
-2
votes
1answer
106 views

Contraction map in extending domain from Dense subset to $L^{2}$

This thread is about extending a dense domain $D \subset L^{2}$ into $L^{2}$. I do not understand what Deyton means in his comment about getting contraction map when doing this. I cannot see any ...
8
votes
4answers
1k views

Dirac delta of nonlinear multivariable arguments

How does one compute a dirac delta function with a multivariable argument? For example, compute: $$ \int^{\infty}_{-\infty}{\rm d}x\,{\rm d}y\, \delta\left(x^{2} + y^{2} - 4\right) \delta\left(\left[...
0
votes
1answer
1k views

integral of Dirac delta function with sine

It i well known that the Dirac Delta Function has the following property $\int_{-\infty}^{\infty}\delta(t-a)f(t)dt=f(a)$ if $g(t)=\int_{0}^{t}\sin(t-\tau)\delta(\tau-\pi)d\tau$ then $g(t) = \left\{...
-2
votes
1answer
90 views

Is $\delta : \mathcal{S}(\mathbf{R}) \to \mathbf{C}$ continuous with usual seminorm?

I am thinking again the accepted answer which is found here: When viewing $\delta: \mathbf{S} \to \mathbf{R}$ (linear and continuous with respect to the usual semi-norms on the Schwartz-space ...
0
votes
2answers
38 views

To write this sentence about a distribution more rigorously

I have the sentence at the moment Notice that for all $c \in \mathbf{C}$ such that $W(cx) = |c|^{2} Wx$. which I do not like. I mean to say that for all $c \in \mathbf C$ the equation is true. ...
1
vote
1answer
88 views

Are these $L^{p,q}$ Lebesgue spaces?

I have a theorem, deduced mostly from Loukas' Classical Fourier Analysis book 2009: Let $(X,\mu)$ be a measure space. Then for all $0 <p,q \leq \infty$, the Lebesgue spaces $L^{p,q}(X,\mu)...
0
votes
2answers
97 views

Does delta distribution remain continuous with respect to quasinorm?

I am thinking the accepted answer which is found here: When viewing $\delta: \mathbb{S} \to \mathbb{R}$ (linear and continuous with respect to the usual semi-norms on the Schwartz-space – or ...