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
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
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1answer
38 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
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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
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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
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2answers
136 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
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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 ...
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0answers
44 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
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1answer
68 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).
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1answer
58 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
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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
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1answer
102 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
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1answer
31 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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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
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4answers
201 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
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1answer
327 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
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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
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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
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2answers
76 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
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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
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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
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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
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2answers
152 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
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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
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0answers
60 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
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0answers
80 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
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2answers
97 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
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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
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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
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1answer
97 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 ...
0
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2answers
107 views

Dirac's delta definition

Wikipedia gives the following definition to Dirac's delta: $$\delta(x-\alpha)=\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ip(x-\alpha)} dp $$ but solving the integral we get: $$\delta(x-\alpha) = ...
7
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4answers
194 views

Iterated Limits Schizophrenia

Consider the functions $g_n(x)$, with $n\in\mathbb{N}$, $n \ge 1$ and $x\in\mathbb{R}$, defined as follows: $$ g_n(x) = \begin{cases} 2n^2x & \text{if }0 \le x < 1/(2n) \\ ...
3
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1answer
62 views

What´s wrong in this computation of $\Delta(r^{-1})$ as a distribution?

maybe this is an idiot question, but I could not figure out what´s wrong. I know how to compute $\Delta (r^{-1})$ in $\mathbb{R}^{3}$ putting a ball with center in $0$ and then get $\Delta(r^{-1}) = ...
0
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1answer
53 views

Differential equation

What method should I follow if I want to solve the equation $u''-u=\delta_0+\delta_1$ in $\mathcal{D}'(\mathbb{R})$ ?? Thanks in advance!
2
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1answer
354 views

Fourier Transform of Dirac Comb on $\mathbb{Z}$ and $\mathbb{Z}^{d}$.

Let $f(x)=\sum_{n\in\mathbb{Z}}\delta(x-n).$ (a) Show $f$ is a tempered distribution. (b) Compute $\hat{f}$ using the convention $\int_{\mathbb{R}}f(x)e^{-ix\xi}\;dx$ convention for $\mathcal{F}$. ...
3
votes
1answer
115 views

If $f\in L^1(\mathbb{R})$ is such that $\int_{\mathbb{R}}f\phi=0$ for all continuous compactly supported $\phi$, then $f\equiv 0$.

I am wondering about a proof of the fact that If $f\in L^1(\mathbb{R})$ is such that $\int_{\mathbb{R}}f\phi=0$ for all continuous compactly supported $\phi$, then $f\equiv 0$. I am familiar with the ...
4
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0answers
73 views

Regularity theorem for Laplacian

Let $\Omega \subset \mathbb R^d$ be a bounded domain, $d>2$. Let $f \in C^\infty(\Omega)$. If $u \in L^2$ is a distributional solution of $\Delta u = f$ in $\Omega$ then $u \in C^\infty(\Omega)$ ...
3
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2answers
78 views

Constructing a Distributional Solution to the Inhomogeneous C.R. Equations

The question is to find a fundamental solution to the system of equations in $\mathbb{R}^{2}$ \begin{array}{l} u_{x}-v_{y}=f\\ u_{y}+v_{x}=g\end{array} and to express the answer as a $2\times2$ ...
2
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1answer
40 views

A distribution $u=\frac{1}{x}$

I am interested in finding a distribution $u \in \mathcal{D}'(\mathbb{R})$ such that $u=0$ on $(-\infty,0)$ and $u=\frac{1}{x}$ on $(0,\infty)$. This is exercise 1.4 in Friedlander. Hints or help ...
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0answers
102 views

Fundamental solution of wave equation in 3D

I want to ask for assistance in verifying the fundamental solution of the wave equation in $\mathbb{R}^{3}$. Here the fundamental solution is given by $$\frac{1}{2\pi}H(t)\delta(t^{2}-|x|^{2})$$which ...
0
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2answers
81 views

What is the closure of $ C^\infty_c(\mathbb{R}^n\setminus\{0\})$ in Sobolev $ W^{1,p} $ norm?

For $1 \leq p < \infty, n\geq 1 $ my guess of the answer was $ W^{1,p}(\mathbb{R}^n)$ but I can't prove the inclusion $ \overline{C^\infty_c(\mathbb{R}^n\setminus\{0\})} \subseteq ...
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0answers
28 views

$u \in H^n$ then also $|u|^2 \in H^n$ for complex valued function in Sobolev space

Suppose that a complex valued function $u$ is in the Sobolev space $H^n(R^n) = W^{n,2}$. Is it necessarily true that $|u|^2 \in H^n$ ? The result is true for real - valued functions since we know ...
0
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1answer
95 views

Convolution of functions and measures

I need some help with this exercise. I'm not sure how to deal with it: Let $f(x)=e^{-x^2}$, $\mu$ the Lebesgue measure in $[0,1]$ and $\nu$ the Lebesgue measure in $[2,\infty)$. I have to find the ...
0
votes
1answer
52 views

Tempered fundamental solutions

According to the Malgrange–Ehrenpreis theorem every nontrivial linear constant coefficient PDO $P(\partial)$ admits a fundamental solution $E\in\mathscr{D}'$; I wonder whether $P(\partial)$ admits a ...
1
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1answer
89 views

Convolution of distributions is not associative

I need some help with this exercise: It proposes to show that convolution of distributions is not associative: If $T=T_1$ (distribution given by f=1), $S=\delta'$, and $R=T_H$ (we denote as $H$ the ...
2
votes
2answers
132 views

How to treat $\int_{0}^{\infty} \sin(kx)dx =\frac {1}{k}$ as a distribution?

How to evaluate the following integral? $\int_{0}^{\infty} \sin(kx)dx=\frac 1 k$ The book Mathematical Physics by Butkov reads "The sequence $f_N(k)=\int_{0}^{N} \sin(kx) dx=\frac{(1-\cos ...
6
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1answer
135 views

What is $\Delta\frac{1}{|\mathbf{x}|^2}$, as a distribution?

$\newcommand{\x}{\mathbf{x}}$Let $\x$ denote a vector in $\mathbb{R}^3$, $|\x|$ its magintude and $\Delta=\frac{\partial^2}{\partial x 2}+\frac{\partial^2}{\partial y 2}+\frac{\partial^2}{\partial z ...
1
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1answer
84 views

Primitive of a distribution

I need some help with this exercise, about calculating the primitive of a distribution $T$ given by a series. Is the following: ...
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0answers
40 views

Neumann boundary condition for smooth function defined on the interior

Let $\Omega\subset\mathbb{R}^n$ be open and let $f\in C^\infty(\Omega)$ be a smooth function. What examples can one come up with that distinguish the 3 criteria below? 1: f satisfies the Neumann ...