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
50 views
Identify the distrionbutional derivative with classical derivative?
I am reading Rudin's Functional Analysis and got quite confused by his proof for theorem 7.25, which he calls Sobolev's Lemma.
In proving the theorem, he defines the function $F$, and calculates its ...
1
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1answer
168 views
Easy question on derivative in the sense of distribution
I would like help proving this elementary result:
Let $f\in L^{1}_{loc}(a,b)$. Let $x_0 \in (a,b)$ Let $F(x)=\int^{x}_{x_0} f$. Then $F'=f$ in the sense of distributions.
i.e How do I show ...
3
votes
2answers
107 views
Howto show that function is a representation fot the delta function via complex path integrals?
So given is the definition:
$$ f(x):=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}e^{ikx}dk $$
I'm supposed to show that this is a representation of the Dirac delta "function" ($f(x) = \delta(x)$) ...
1
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1answer
192 views
Delta Dirac Function
Show that $\delta$, function of Dirac, defined than $\left<{\delta_0,\phi}\right> = \phi(0)$ belongs to $W^{-1,p}(]-1,1[)$ and $\delta_0 \notin L^p(-1,1)$ $\forall p\geq 1$.
How I will be able ...
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2answers
77 views
Why begin with distributions and then move to tempered ones?
After reading several books on distribution theory, I got a strange feeling. Why do they all begin with the theory of distributions and then move on to tempered distributions? Why can't we just start ...
5
votes
2answers
129 views
Paley-Wiener type theorems for distributions?
In general a theorem of Paley-Wiener type gives a relation between the decay of a function and the smoothness of its Fourier transformation, and there are plenty of them since there are many kinds of ...
1
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1answer
96 views
If $f$ is a bounded tempered distribution and $g \in L^1$ is then $\int_{\Bbb R^n}(f\ast\tilde\varphi)(x)\tilde g(x)\,dx$ a tempered distribution?
Let $f$ be a bounded tempered distribution, that is, $f\ast\varphi \in L^\infty(\mathbb R^n) $ for every Schwartz function $\varphi$. If $g \in L^1(\mathbb R^n)$, does the following definition define ...
2
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1answer
45 views
Laplacian in $\Bbb R^2$ acting on compact test-function
I am trying to follow an argument in Strichartz's "A Guide to Distribution Theory and Fourier Transforms"
We consider $\langle \Delta u, \rho \rangle$ where $\Delta u$ is the two dimensional ...
1
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1answer
121 views
What is good about homogeneous functions?
Given $r>0$ and $f:\mathbb{R}^n\to \mathbb{R}$, $d_rf$ is the function defined by \begin{equation}d_rf(x_1,x_2,\dots,x_n)=f(rx_1,rx_2,\dots,rx_n)\end{equation} and is called the $r$-dilation of ...
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2answers
104 views
How to cook up test functions?
Let $\Omega\subset \mathbb{R}$ be open. A test function is a $C^\infty$ function with compact support. This is a rather strong restriction, for instance, no analytic function is a test function. But ...
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0answers
78 views
What is $\overline{\partial} 1/z^2$?
it is all in the title : what is $\overline{\partial} \frac{1}{z^2}$ in the sense of distributions ? I remember that $\overline{\partial} \frac{1}{z}$ is a dirac at 0, but I can't seem to find a way ...
5
votes
1answer
170 views
Fractional derivatives of delta function $ \delta (x) $
How can I define the fractional derivative of the Delta function?
I mean $D^{\alpha}= \frac{d^{\alpha}}{dx^{\alpha}} $ where $\alpha$ can be any real number, then if we define $D^{\alpha} \delta (x) ...
2
votes
1answer
95 views
generalized functions (Distributions) elementary question
I am working with Strichartz's "A Guide to Distribution Theory and Fourier Transforms" (self-study -> not a homework question). He says none of the distributions that correspond to $1/|x|$ are ...
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1answer
111 views
Confused by a proof in Rudin's Functional Analysis
I am referring to a proof in Part II of Rudin's Functional Analysis.
I got confused by his proof of Thm 6.26 (page 167). He says by applying (2) successively we can get inequality (4), but I do not ...
1
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1answer
51 views
Nice example where $D^{\alpha}\Lambda_{f}\neq\Lambda_{D^{\alpha}f}$?
Let $\Omega\subset\mathbb{R}^n$ be open and $f$ be a locally integrable function. The distribution associated with $f$, $\Lambda_{f}\in D'(\Omega)$, is defined via \begin{equation}
...
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0answers
23 views
Notation Issues
I am reading a paper, and have come across a notation I don't understand, it says:
To the resulting sequence of orthonormal eigenfunctions we may associate a sequence of distributions ${dU_{k_i}}$ in ...
2
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0answers
47 views
Good references on Distribution Theory [duplicate]
Possible Duplicate:
Distribution theory book
Two books I have been reading are Strichartz's A Guide to Distribution Theory and Fourier Transforms and PartII of Rudin's Functional Analysis . ...
2
votes
0answers
83 views
Liouville's Theorem in $\mathbb{R}^n$
Liouville's Theorem states that if a tempered distribution is harmonic, $\Delta{u}=0$, then $u$ is given by a polynomial. For the argument, we take Fourier transform of $\Delta{u}=0$ to obtain ...
4
votes
2answers
174 views
Convergence of test-functions is not induced by any metric.
By $\mathcal{D}(\mathbb{R})$ we denote linear space of smooth compactly supported functions. We say that $\{\varphi_n:n\in\mathbb{N}\}\subset\mathcal{D}(\mathbb{R})$ converges to ...
1
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1answer
102 views
Delta function question
Given the functions
$$f(x)= \delta (x-a)$$
$$g(x)= \frac{1}{a} \delta \left(x- \frac{1}{a}\right)$$
for a real constant $a\gt0$, is there a relationship between $f$ and $g$?
I believe that $ ...
5
votes
2answers
219 views
Square root of compactly supported C-infinity function
Given $u \in \mathcal{C}^\infty_0(\mathbb{R}^n)$, $u \geq 0$ everywhere, is $v(x) = \sqrt{u(x)}$ also in $\mathcal{C}^\infty_0$? It is clear that the only problematic points are the boundary of the ...
2
votes
2answers
143 views
Regarding the definition of Schwartz Space of functions
I came across a definition of Schwartz Space where they were defined as functions $f$ such that $\mathrm{lim}_{|x|\to \infty} |x^{\alpha}D^{\beta}f(x)|=0$ for any pair of multiindices $\alpha,\ ...
0
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1answer
77 views
how to compute the convolution of two measures explicitly
Here is my example:u and v are the surface measures on the spheres {${x;|x|=a}$} and {${x;|x|=b}$} in $\mathbb{R}^{3}$.Then what's $u\ast v$ ? And what if in $\mathbb{R}^{n}$?
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0answers
80 views
What's the Fourier transform of these functions?
The Fourier transform of $|x|^{\alpha}$.
This is the Fourier transform of a homogeneous function, and there are several cases of various $\alpha$: when $a\leq -n$, it's not a temperate distribution; ...
4
votes
1answer
126 views
a question about convolution of two distributions
Generally,when taking convolution of two distributions,at least one of which is supposed to be of compact support.
But when u,$v\in S'(\mathbb{R})$ ( temperate distributions) have suports on the ...
2
votes
1answer
78 views
Some questions about distribution theorem
Given an equation $P(D)u=0$, where $P$ is a polynomial (not equal to a constant). Here are some basic information about the distributional solution $u$:
If $P$ has at least one real root, then there ...
1
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1answer
367 views
Proving the mean value property of harmonic functions using distributions?
A professor I talked to showed me a proof of the mean value property. (He actually showed it for functions solving the heat equation instead of Laplace's equation, but it seems like the argument is ...
3
votes
1answer
179 views
The distribution $\Delta u$ (where $u = \ln|\vec{x}|$)
Problem
Consider the function $u(\vec{x})=\ln|\vec{x}|$ as a distribution on $\mathbb{R}^3$ and $\mathbb{R}^2$. We want to determine $\Delta u$ in the distribution sense. First calculate $\Delta u$ ...
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1answer
161 views
What is the sum of only half the exponential terms that give the Dirac comb?
The following infinite sum of exponential terms gives a Dirac comb:
$$ \sum_{n=-\infty}^\infty e^{i n x} = 2 \pi \sum_{n=-\infty}^\infty \delta(x - 2 \pi n) $$
Of course the sum doesn't strictly ...
2
votes
1answer
231 views
Normalization parameter, properties of Dirac delta functions
Suppose $\psi_E (x)=N(E)\exp (ikx)$
where $\psi_E (x)$ is a momentum eigenfunction, $N(E)$ is the normalization constant on the energy scale such that $\langle E'|E\rangle=\int_{-\infty}^\infty ...
0
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0answers
106 views
Integration methods for functions with Delta distributions
Which Monte-Carlo methods are available for computing a multidimensional integral with Delta distributions (in case one cannot sample them explicitly)?
PS: I also asked a similar question at ...
3
votes
0answers
73 views
Colombeau product of distributions
How can use the Colombeau generalized function method to evaluate the product of distributions
$ \delta (x) \times \delta (x) $ or $ \delta ^{n} (x) \times \delta ^{m} (x) $ (derivatives of dirac ...
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votes
0answers
49 views
Why we have “$\Delta \Gamma = \delta$”, in $\mathbb{R}^{n}$, where $\Gamma$ is fundamental solution and $\delta$ is the dirac measure?
Why we have "$\Delta \Gamma = \delta$", in $\mathbb{R}^{n}$, where $\Gamma$ is fundamental solution and $\delta$ is the dirac measure?
4
votes
2answers
241 views
limit of an integral with a Lorentzian function
We want to calculate the $\lim_{\epsilon \to 0} \int_{-\infty}^{\infty} \frac{f(x)}{x^2 + \epsilon^2} dx $ for a function $f(x)$ such that $f(0)=0$. We are physicist, so the function $f(x)$ is smooth ...
3
votes
1answer
45 views
Uniform convergence and convergence in $S'(\mathbb{R}^n)$
Let
$$\hat{f_\epsilon}: \xi \mapsto \exp(-\epsilon |\xi|) \frac{\sin(|\xi|t)}{|\xi| t}$$
denote to the Fourier transform of $f$. How do I see
$\hat{f_\epsilon}$ converges uniformly on ...
-1
votes
1answer
117 views
Approximating an integral
Might be simple, but i don't get it. Why is the integral in the last line approximately equal to $n(\varphi(\frac{-1}{2n}) - \varphi(\frac{1}{2n}))$?
4
votes
1answer
231 views
Application of closed graph theorem.
I'm having a problem applying the closed graph theorem, which I think stems from distributions still being very new to me.
I am reading a proof in Stein and Weiss, Introduction to Fourier Analysis ...
3
votes
0answers
54 views
Why $\partial_{i}(A^{*}u)=A^{*}(\sum^{n}_{j=1}a_{ji}\partial_{j} u)$?
We define the affine transformation on distributions by $$\langle A^{*}u, \phi \rangle=\frac{1}{\det(A)}\langle u,\phi(A^{-1}x)\rangle$$
Assume this we should have $$\langle \partial_{i}(A^{*}u), ...
1
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0answers
47 views
wavefront set of a distribution
If $(x_0,\xi_0)\in\mathbb{R}^{2n}$ is a given point in phase space, how do I construct a compactly supported distribution $u$ which has
WF$(u)=\{(x_0,t\xi_0) | t>0\}$ ?
0
votes
1answer
334 views
Fourier transformation of sin, cos, sinh and cosh
I am trying to solve the following exercise
Use $\mathcal{F}(e^{xb}) = 2\pi \delta_{ib}$ to calculate the Fourier-Transformation of $\sin x$, $\cos x$, $\sinh x$ and $\cosh x$
Now I am a little ...
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0answers
119 views
Poisson equation on half-space
Let $H$ be the open half-space of $\Bbb R^n$ defined by $x_n > 0$.
Let $f : \overline H \to \Bbb R$ continuous and harmonic on $H$.
Define the function $F : \Bbb R^n \to \Bbb R$ by
$$ ...
0
votes
1answer
78 views
How to use the Malgrange-Ehrenpreis-Theorem
In the Wikipedia article of this theorem
http://en.wikipedia.org/wiki/Malgrange%E2%80%93Ehrenpreis_theorem
it is said that i could be used to prove that
$P(\partial/\partial x_i)u(x)=f(x)$
has a ...
4
votes
1answer
134 views
Questions concerning a proof that $\mathcal{D}$ is dense in $\mathcal{S}$.
I am currently working through
this lecture notes
and on page 164, there it is said
The space of $\mathcal{D}(\mathbb{R}^n)$ of smooth complex-valued functions with compact support is contained ...
2
votes
0answers
68 views
Why does the following define a distribution and of which order?
I want to show that
$$\phi\mapsto\underset{\varepsilon\searrow 0}{lim}\int_{-\infty}^{\infty}\frac{\phi(x)}{x+i\varepsilon}dx$$
defines a distribution on $\mathcal{D}(\mathbb{R})$ but I just don't ...
2
votes
1answer
239 views
Does zero distributional derivative imply constant function?
If a real function $f\colon[a,b]\to\mathbb{R}$ is differentiable and its derivative $f'$ is zero, then $f$ is constant. Does this result still hold when $f$ has a weak derivative?
Explicitly, suppose ...
1
vote
1answer
45 views
whats the order of a distributional derivate?
I have to calculate the derivatives of order $\le 2$ of for example $f(x) = |x|$, is it the same as the second derivate, what does this "of order $\le 2$" mean? calculating distributionell derivatives ...
1
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1answer
36 views
connection between the support and the representation of a distribution
I want to show, that for $u' \in \mathcal{D}'(\mathbb{R}^n)$
supp $u$ = $\{ 0 \}$ iff there exist numbers $m \in \mathbb{N}, c_{\alpha} \in \mathbb{K}$ such that $u = \sum_{|\alpha| \le m} c_{\alpha} ...
0
votes
2answers
65 views
why is $\frac{\phi(x)-\phi(-x)}{x}$ for smooth $\phi$ bounded at $x=0$
Why is $\frac{\phi(x)-\phi(-x)}{x}$ for smooth $\phi$ bounded at $x=0$? If i set $\phi(x) = \sqrt{|x|}$, it definitely not bounded. I saw this on page 293 of
...
2
votes
2answers
215 views
principal value as distribution, written as integral over singularity
Let $C_0^\infty(\mathbb{R})$ be the set of smooth functions with compact support on the real line $\mathbb{R}.$ Then, the map
$$\operatorname{p.\!v.}\left(\frac{1}{x}\right)\,: ...
2
votes
1answer
73 views
what means a integral exists in the distributional sense?
what exactly means if an integral exists just in the distributional sense, for example the fourier-transform of $x^2 e^{-\lambda x}$ or of $H(R-|x|)$ where $R > 0$ and $H$ is the ...
