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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1answer
48 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
51 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
52 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
65 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
40 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
159 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
57 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
46 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
98 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
185 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
37 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
61 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
67 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
112 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
90 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
34 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
60 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
18 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
140 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
59 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
45 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
71 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
59 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
37 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 ...
1
vote
1answer
103 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
89 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
32 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 ...
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votes
1answer
75 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 ...
6
votes
4answers
211 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) ...
0
votes
1answer
342 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) = ...
-2
votes
1answer
70 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
31 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
59 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 ...
0
votes
2answers
78 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 ...
0
votes
0answers
77 views

Contraction mapping and $L^{2}(\Bbb R \times \Bbb R)$ spaces in inequality

I found this: \begin{equation} \lVert Wx \rVert^{2}_{L^{2}( \mathbb{R} \times \mathbb{R} )} \leq \lVert x \rVert^{2}_{L^{2}(\mathbb{R} )}, \end{equation} which I think ...
7
votes
0answers
94 views

In the space of distributions, how big is the subspace of functions?

I'm teaching Distribution theory and many of my students still believes that there is only one kind of distribution : the distribution that can be identified to a $L^1_{\text{loc}}$ function. And I ...
0
votes
1answer
33 views

Differentiability of smooth functions

The space is from Schwartz space $S$ or the space of all smooth functions of compact support $D$. Can you say anything about how many times the test function is differentiable from this? I think ...
0
votes
2answers
155 views

Discontinuity of Dirac Delta distribution

I know that the following holds for Step function, but not sure if it holds for the distribution too. Does the following hold for the Dirac delta distribution too? $\delta$ is a linear functional ...
2
votes
0answers
36 views

Multiplying and dividing distributions by non-$C^\infty$ functions.

It's quite easy to see that we can multiply distributions by any $\mathcal C^\infty $ functions. Moreover, if the distribution $T$ is of order $k$, then we can mupliply it by a $\mathcal C^k$ ...
3
votes
0answers
62 views

why is test function space $\mathcal{A}$ complete

I am trying to find out, why the space $$\mathcal{A}:=\left\{\phi\in C_0(\mathbb{R}^{2d})|\;\|\phi\|_\mathcal{A}:=\int_{\mathbb{R}^d}\sup_{x\in\mathbb{R}^d}|(\mathcal{F}_p\phi)(x,y)|\;\mathrm ...
8
votes
0answers
81 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
2answers
103 views

About a Fourier transform of a non- integrable function.

I'm trying to obtain the Fourier transform of the following function: $$F(x)=\frac{x}{1+x^2}$$ I have tried using Residue Theorem, but i think it can't be applied because the difference between the ...
1
vote
1answer
77 views

About convolution and Fourier transform

I have some doubts with this question: I we have $f,g\in\cal{S}$ (where $\cal{S}$ is the Schartz space) with $f\ast g=0$, Can we deduce that $f=0$ or $g=0$? What I did is apply Fourier transform, ...
1
vote
1answer
54 views

About the k-th derivative of the Delta function

I need some help to compute the k-th derivative of the Dirac's Delta function, $\delta_0^{(k)}$. I know its Fourier transform is $TF(\delta_0^{(k)})(y)=(iy)^{k}$( I don't know if this could be ...
1
vote
1answer
118 views

About the Fourier transform of the sign function

I'm trying to calculate the Fourier transform of the function $f(x):=sign(x)$. I have read some texts where this is solved approximating the function $f$ by other functions, $f_a$, defined as follows ...