3
votes
0answers
26 views

Hopf lemma for generalized normal derivatives

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. If $u\in C^2(\Omega)\cap C_0(\overline{\Omega})$ is a superharmonic function ($-\Delta u\ge 0)$ then, Hopf Boundary Lemma does implies that ...
2
votes
0answers
57 views

Is this distributional laplacean a measure?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Suppose that $0<k<t<1$ and $\Phi$ is a mooth function satisfying: $\Phi(x)=0$ for $x\le k$, $\Phi(x)=1$ for $x\ge t$. Take $u\in ...
2
votes
1answer
48 views

Can we conclude that $\Delta (\Phi\circ u)$ is a measure, given that $\Phi$ is a particular smooth function and $u$ is in some Sobolev space?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Assume that $u\in W_0^{1,2}(\Omega)$ is such that $\Delta u$ is a measure. Let $\Phi$ be a smooth function in $\mathbb{R}$, such that ...
2
votes
1answer
96 views

Dirac delta distribution and measure?

Of course the Dirac delta is not a function. Despite, I think the concept of a measure is much easier than that of a distribution. Therefore, I was wondering: In what sense is the concept of a Dirac ...
1
vote
1answer
189 views

The issue of treating an inverse Fourier transform in terms of a tempered distribution.

Consider the wave equation $$ u_{tt}=\Delta{u} \quad u(x,0)=f(x) \quad u_t(x,0)=g(x) \tag{*} $$ A solution to this equation is given by $$ u(.,t)=f*\partial_t\Phi_t+g*\Phi_t \tag{**} $$ where ...
0
votes
1answer
46 views

Computing Fourier transform of a surface measure.

I am almost beginner in the topics of Fourier transform. So, I am asking this question here. Let $n=3$ and let $\mu_t$ denote surface measure on the sphere $|x|=t$. Then how do we show that $$ ...
1
vote
1answer
35 views

Examples of functions with values in distributions

What is an example of a function in $L^p((0,T);\mathcal{D}'(\mathcal{R}))$? I ask this because the Majda-Bertozzi book on Incompressible flow deals with vortex sheet initial data $\omega(t,\cdot)\in ...
1
vote
1answer
34 views

Expected distance within a distribution is smaller?

consider we have two general distributions $f_1$ and $f_2$, assume they have different support $S_1$ and $S_2$. Is the expected distance btween two points draw from the same distribution smaller than ...
0
votes
0answers
30 views

Fourier transform of a function of characteristic function of a measure

Let $\mu$ be complex measure on $\mathbb{R}^2$ ($|\mu|$ is finite measure) and $\chi$ - its characteristic function $$ \chi(x_1,x_2) = \int_{\mathbb{R}^2} d\mu(p_1,p_2) \exp(i p_1 x_1+i p_2 x_2). $$ ...
6
votes
1answer
75 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 ...
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 ...
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 ...
2
votes
1answer
118 views

Convergence of a integral - heat Kernel and dirac delta function

Consider $\varphi \in S(R^n)$ (space of rapidly decreasing functions). Consider the heat kernel $$ K_t(x) = \displaystyle\frac{1}{{(4\pi t)}^{n/2}} \displaystyle e^{- \displaystyle\frac{|x|^2}{4t}}, ...
-1
votes
2answers
91 views

distributional Laplacean of a function and the dirac delta distribution

Consider $S(R^2)$ the space of rapidly decreasing functions (http://en.wikipedia.org/wiki/Schwartz_space). Consider $F(x) = \displaystyle\frac{1}{2 \pi} \ln|x| , x \in R^2 - \{ 0 \}$. I want to ...
5
votes
0answers
164 views

If a Radon measure is a tempered distribution, does it integrate all Schwartz functions?

The question might at first sight sound like the answer is trivially "yes", so let me clarify the question a bit. Consider given a Radon measure $\mu$ on $\mathbb{R}^n$. Let ...
2
votes
1answer
69 views

Correct notation when integrating Dirac distribution

I have a question regarding the correct notation when integrating the Dirac distribution $\mu$. When treating it as a measure, I can write the Lebesgue inetgral $\int_{\mathbb{R}}\mu(dx)=1.$ What if I ...
5
votes
0answers
83 views

Show that integrals are equal

Let $f \colon [0,+\infty) \to \mathbb R$ be a convex function. Then $f''(x)$ is a nonnegative distribution on $(0,+\infty)$ and hence it can be continued to a nonnegative mesure $\mu$ on ...
4
votes
1answer
133 views

Compatibility of pointwise and distributional convergence

This has probably been asked before but I couldn't find it. Let $\Omega$ be an open subset of $\mathbb{R}^n$ and let $u_k,\, u \in L^1_{\mathrm{loc}}(\Omega)$ and $v\in \mathscr{D}'(\Omega)$ (the ...
3
votes
1answer
109 views

Gauss–Ostrogradsky formula for Distributions

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $u\in W^{1,p}(\Omega)$, $v\in L^{p'}(\Omega)^N$ with $p\in (1,\infty)$. Let $\operatorname{div}(v)$ be the divergence of $v$ in the sense of ...
0
votes
1answer
91 views

A basic question about $\operatorname{supp}f$ (support of f).

Is it true that $\operatorname{supp}f$ is the complement of the biggest open set where $f=0 $? Here $\operatorname{supp}f=$ {$x\in \Bbb R^n ; f(x)\not=0$} and $f\in C$ (collection of continuous maps ...
0
votes
1answer
67 views

A clarification about BV functions.

From definition, a locally integrable $ u \in BV(\Omega) $ if its distribution derivative is given by a signed Radon measure. That is there exists $ \mu $ such that for any $ \phi \in ...
9
votes
2answers
138 views

Various kinds of derivatives

Let $f\colon \mathbb{R}\to \mathbb{R}$ be a measurable function. Let us introduce the following notions of "derivative" of $f$. Classical derivative. The unique function $f'_c$ defined pointwise by ...
1
vote
0answers
62 views

Using $\frac{1}{A+i\epsilon} = PV\frac{1}{A}-i\pi\delta(A)$ in Feynman Integrals

Is the following operations OK (this is related to the Feynman parameter trick)? $$F:= \int_0^1 \mathrm{d}x\int_0^{1-x}\mathrm{d}y \frac{1}{f(x,y)+\mathrm{i}\epsilon}.$$ Now using ...
0
votes
1answer
220 views

Questions related to distribution function and its “inverse”

Let $f: \mathbb R^n \to \mathbb R$ be a measurable fucntion. Define $F(t) = \mu \{x:|f(x)| >t\}$ Show that $F$ is nonincreasing and right-continuous (done). Define $F^\star(v)=\inf \{t: F(t)\leq ...
2
votes
0answers
51 views

Representation of distribution by nonnegative measure

Let $T \in \mathcal{D}'(\mathbb{R}_{+})$ be a distribution on $\mathbb{R}_{+}$ such that for any $f \in \mathcal{D}(\mathbb{R}_{+})$, $f \geqslant 0$ we have $$ \langle f, T \rangle \geqslant 0 $$ ...
1
vote
0answers
28 views

Solve an equality for distribution

I have an equality that holds for any $\lambda > 0$ $$ \int\limits_{0}^{\infty}{e^{-\lambda t^{\alpha}}} T(t)dt = \int\limits_{0}^{\infty}e^{-\lambda t^{\alpha}} dg(t), $$ where $\alpha > 0$ ...
2
votes
0answers
54 views

Action of linear functional on integral depending of parameter

Let $K(x,\omega) \in C^{\infty}(\Omega \times \Omega)$, where $\Omega$ is a domain in $\mathbb{R}^{n}$. Let $\mu$ be a probability measure on $\Omega$. My question is under which conditions an ...
5
votes
1answer
312 views

How should I understand a PDE that contains distribution or measure mathematically?

We know that the Dirac delta can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite, $$ \delta(x)=\begin{cases}+\infty, ...
2
votes
2answers
367 views

Convolution of different objects

I am studying an article in which convolution is used very much and it is used between all sorts of objects: functions, distributions and measures. I know that convolving a $L^1$ function (for ...
4
votes
3answers
242 views

Borel Measure such that integrating a polynomial yields the derivative at a point

Does there exist a signed regular Borel measure such that $$ \int_0^1 p(x) d\mu(x) = p'(0) $$ for all polynomials of at most degree $N$ for some fixed $N$. This seems similar to a Dirac measure ...
3
votes
1answer
310 views

The limit of locally integrable functions

If ${f_i} \in L_{\rm loc}^1(\Omega )$ with $\Omega $ an open set in ${\mathbb R^n}$ , and ${f_i}$ are uniformly bounded in ${L^1}$ for every compact set, is it necessarily true that there is a ...
9
votes
2answers
213 views

How do different notions of “distribution” relate to one another?

In reading "Real Analysis: Modern Techniques and Their Applications" (Folland), I've come across a few different notions of "distribution" or "distribution functions." The distribution function of a ...