# Tagged Questions

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).

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 ...
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 ...
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" ($f(x) = \delta(x)$) ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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) ...
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 ...
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 ...
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} ...
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 ...
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 . ...
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 ...
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 ...
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 \$ ...
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 ...
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,\ ...
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 $\mathbb{R}^{n}$?
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; ...
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 ...
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 ...
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 ...
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 $\Delta u$ ...
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 ...
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 ...
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 ...
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 ...
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?
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 ...
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 ...
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 $n(\varphi(\frac{-1}{2n}) - \varphi(\frac{1}{2n}))$?
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 ...
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), ...
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$(u)=\{(x_0,t\xi_0) | t>0\}$ ?
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 ...
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  ...
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 ...
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 ...
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 ...
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 ...
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 ...
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} ...
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 ...
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)\,: ...
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 ...
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