1
vote
0answers
28 views

Rapidly Decreasing Functions

Can someone explain the notion of a rapidly decreasing function? Namely, a function in the Schwartz space: $$\mathscr{S}(\mathbb{R}^n):= \{ f \in C^{\infty} (\mathbb{R}^n) : ||f||_{\alpha, \beta} ...
0
votes
1answer
22 views

Understanding Distributional Meanings and Test Functions for PDEs

thank you for taking the time to read my question. My question is about distributional meanings in PDEs. My specific question is at the bottom, but I'd be interested in a bit of general theory (even ...
3
votes
1answer
35 views

When does the regularization of a function converges to the function?

Let $\theta(x)$ equal $k\exp(-\frac{1}{1-||x||} )$ if $||x||<1$, and equal 0 if $||x||\geq1.$ Here $||.||$ designates the Euclidian norm in $\mathbb{R}^{^{n}}$, and the constant $k$ is chosen such ...
1
vote
2answers
52 views

Example of a smooth 'step'-function that is constant below 0 and constant above 1

I need an infinitely smooth non-decreasing function $\ f(x)$, that $$f(x)=0\quad\forall x\leq 0,$$ $$f(x)=1\quad\forall x\geq 1,$$ and all its derivatives in $x=0$ and $x=1$ are $0$. I found that I ...
4
votes
1answer
42 views

How to show distribution $T$ is a constant

Let $T$ be a distribution on $\mathbb{R}$. $\tau_a\phi(x)=\phi(x-a)$ and $\langle T,\tau_a\phi\rangle=\langle T,\phi\rangle$ for all $a\in \mathbb{R}$ and all test functions $\phi$. Prove that $T$ ...
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 ...
2
votes
1answer
45 views

Weak-Strong Derivatives

If a continuous function $u:\mathbb R^d\to \mathbb R$ has a weak derivative $\frac{\partial u}{\partial x_j}$ that exists everywhere as a locally integrable function, and it is even continuous, does ...
1
vote
0answers
37 views

Does the fundamental theorem of calculus hold for BV functions?

I am a bit confused and I hope you can help me in understanding a bit better these things. Let us start by considering one dimensional case. Let $f\colon \mathbb (a,b) \to \mathbb R$ be a function. ...
0
votes
0answers
117 views

Delta function?

I came across this integral: \begin{align} \int_{-\infty}^{\infty} \mathrm{d}k_1 |f(k_1)| \int_{-\infty}^{\infty} \mathrm{d} k_2 |g(k_2)|& \nonumber \times\bigg\{\lim _{V \to ...
0
votes
1answer
19 views

Close fourier transforms implies close time domain functions?

What conditions do we need so that $A(f)\approx B(f) \Rightarrow \mathcal{F}^{-1}\left\{A\right\}(t)\approx\mathcal{F}^{-1}\left\{B\right\}(t)$ The Fourier transforms of my two things look alike. In ...
1
vote
0answers
55 views

Weak convergence implies uniform convergence in distribution?

Consider an empirical processes $\{v_T(\theta), T\geq 1 \}$ and assume it is weakly convergent to the stochastic process $\{ v(\theta); \theta \in \Theta\}$., i.e. $$ E^*(f(v_T(\theta)))\rightarrow ...
2
votes
1answer
52 views

the continuous functions with norm

I'm having trouble trying to understand what does means the first expression in particular the last term in it should we add $\|f\|_{\infty} \leq \infty$ or what i can't see what is his role ...
3
votes
1answer
108 views

How to build a compact support for a function

I was wondering if it is possible to build a distribution with compact support from a function. More precisely, consider a compact set $\mathbf{K}\subset\mathbb{R}^2\setminus\{0\}$, and a function ...
1
vote
2answers
72 views

Derivatives of $|x|$

I wanted to calculate the first and second derivatives of the function $f(x)=|x|$, in order to verify that: $$ f'(x)=\frac{|x|}{x} $$ and $$ f''(x)=2\delta(x). $$ Can you help me?
7
votes
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) \\ ...
0
votes
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 ...
5
votes
0answers
163 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 ...
1
vote
3answers
105 views

Convergence of $\frac{1}{\sqrt{2\pi i\varepsilon}}\exp\left( -\frac{x^2}{2i\varepsilon}\right)$ as $\varepsilon \to 0$

Is it true that $\frac{1}{\sqrt{2\pi i\varepsilon}}\exp\left( -\frac{x^2}{2i\varepsilon}\right)$ converges (in some sense) to $\delta_0$, Dirac delta distribution at point $0$, as $\varepsilon \to 0$ ...
0
votes
0answers
32 views

On the evaluation of a distribution

In $\mathbb{R}^2\setminus\{(0,0)\}$, let $\theta(x,y)$ denote the branch of polar angle satisfying $-\pi<\theta(x,y)\leq \pi$. Since $\theta \in L^1_{loc}(\mathbb{R}^2)$, it can be regarded as a ...
1
vote
1answer
41 views

On showing a distribution is a function

Consider the distributional equation $$\Delta \omega-\omega=\mu$$ Then it is easy to verify by Fourier transform that $$\omega=-\mathcal{F}^{-1}\left(\frac{1}{|\cdot|^2+1}\hat{\mu}\right)$$ is the ...
7
votes
1answer
133 views

Fourier transform using principal value

Can anyone help me compute the Fourier transform of $ 1/|x|^{n-\alpha} $ in $\mathbb{R}^n $ where $ 0 < \alpha < n $ ? Somehow it becomes the principal value of $ 1/|x|^\alpha $ which I can't ...
3
votes
1answer
118 views

How to compute this distribution?

My question refers to this answer. I was hoping someone could explain in more detail the following reasoning. It remains to observe that $\Delta v$ is the distribution composed of the ...
6
votes
1answer
99 views

Uniqueness of Ordinary Differential Equations in $D^{'}$, the space of Schwartz distribuitions

Let $m \in \mathbb{N}$. For $k=1,...,m$ let $a_k : \mathbb{R} \rightarrow \mathbb{C}$ be a $C^{\infty}$ function. And suppose that: $a_m(x) \neq 0 \; \forall x \in [x_0, \infty[$ And let P be the ...
4
votes
1answer
168 views

Local integrability of the convolution of a function with a distribuition

Let $G_n$ be the following distribuitions for $n\geq3$ (for $n=2$ it is just a function) in $\mathbb{R^n}$ (the fundamental solutions of the Laplace equation in $\mathbb{R^n}$ ): ...
2
votes
1answer
82 views

Multiplication in $\mathcal D'(R)$.

I was reading in my text book that multiplication of elements of $\mathcal D'(R)$ is not a continuous function although it is defined and can be seen from the convergence of $\sin(nx)$ to $0$ as $n$ ...
11
votes
3answers
502 views

Dirac Delta or Dirac delta function?

Is Dirac delta a function? What is its contribution to analysis? What I know about it: It is infinite at 0 and 0 everywhere else. Its integration is 1 and I know how does it come.
1
vote
2answers
42 views

Find triple functions $ (g_0,g_1,g_2)$ such that $g_0+g_1'+g_2'' = \delta_0-\delta_1$

I want to find a triple of compactly supported continuous functions $ (g_0,g_1,g_2)$ on $\mathbb{R}$ such that $$g_0+g_1'+g_2'' = \delta_0-\delta_1$$ This is seemingly not so hard but ive broken my ...
1
vote
1answer
55 views

Estimate derivatives in terms of derivatives of the Fourier transform.

Let us suppose that $f: \mathbb{R}^n \to \mathbb{R}$ is a smooth function. Furthermore, for every $\alpha$ multi-index, there exists $C_\alpha > 0$ such that $$ |D^\alpha f(\xi)| \leq ...
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 ...
2
votes
2answers
104 views

Show existence of a continuous $k$ on $\mathbb{R}^2$ such that $(u,\phi)= \int_{\mathbb{R}^2}k\phi dx$ for all $\phi$

(b). Let $u$ be a distribution on $\mathbb{R}^2$. Assume there exists a continuous function $h$ on $\mathbb{R}^2$ such that $(u,\Delta \phi) = \int_{\mathbb{R}^2}h\phi dx $ for all $\phi\in ...
2
votes
1answer
62 views

Showing that $H_{\text{loc}}^2(\mathbb{R}^2) \subset C^0(\mathbb{R}^2)$

I want to show that $H_{\text{loc}}^2(\mathbb{R}^2) \subset C^0(\mathbb{R}^2)$ $C^0$ is the space of continuous functions, and $H_{\text{loc}}^2(\mathbb{R}^2)$ the set of distributions $u\in ...
3
votes
0answers
122 views

Computing the Fourier transform (in the sense of tempered distributions) from a sequence of samples

Let $f\in L_{\operatorname{loc}}^2 (\mathbb{R})$ be a function of period $2\pi$. I'd like to get some help proving the following identity: $$\hat{f}=\sum_{n=-\infty}^{\infty} \hat{f}(n)\delta_n$$ ...
1
vote
2answers
84 views

Show existence of continuous functions $f$ with $f''=\delta_0-\delta_1$

Let $u$ be the distribution on $\mathbb{R}$ given by $$u=\delta_0-\delta_1 $$ (a) show there exists a continuous function $f$ such that $f''=u$ and indicate such one. I thought of doing this with ...
0
votes
1answer
56 views

Operation on distributions

I'm currently studying a course on Advanced Real Analysis for a master degree, and our professor handed to everyone of us a 40-page book. I'm major in Algebra, so I'm not really comfortable with this ...
2
votes
0answers
93 views

Inversion formula for Schwartz-space $\mathcal{S}$.

Let $T\phi = \overline{\mathcal{F}}\mathcal{F}\phi$ for $\phi \in \mathcal{S}(\mathbb{R}^n)$ (rapidly decreasing functions or Schwartz-functions, $\mathcal{F}$ fourier transform and ...
0
votes
1answer
195 views

Is it a dirac-delta?

Hoi, consider $\displaystyle u= \frac{1}{|x|}e^{-|x|}$ for $x\in \mathbb{R}^3$, then one can see that $\Delta u = u$ for $|x|>0$ ( which one can see by transferring $u$ to spherical coordinates). ...
1
vote
1answer
313 views

Inverse fourier transform 3 dimensions

Hoi, I want to show that for $n=3$ that $$\mathcal{F}^{-1}\left(\frac{1}{1+|s|^2}\right) = \frac{1}{4\pi |x|}e^{-|x|} $$ As a hint I've been given: Its the unique solution to the equation ...
9
votes
2answers
137 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
1answer
107 views

Calculation of the Laplacian of a function in $\mathbb{R}^3$.

I have to calculate the Laplacian distributional sense) of the following function $$\frac{e^{i\sqrt{\lambda}|x|}-1}{4\pi|x|}$$ with $\lambda>0$, $x\in\mathbb{R}^3$. I've procedded in the following ...
0
votes
1answer
38 views

Showing $H^1(I) \subset C^{1/2}(\overline{I}) $

Hoi, let $H_1(I)$ the sobolev space on the interval $I = (\alpha,\beta)\subset \mathbb{R}$ http://en.wikipedia.org/wiki/Sobolev_space See here for more on sobolev spaces. $H_m(I)$ contains all ...
0
votes
1answer
40 views

Boundedness of operator

I want to show that the following Linear operator $L$ is bounded and surjective: $L: H^2(I) \to \mathbb{C}^4$ where $I = [\alpha,\beta]\subset \mathbb{R}$ given by $L(u) = ...
0
votes
1answer
98 views

Is this integral 0?

Let $\phi \in C_0^{\infty}(\Omega)$ with $\Omega = (0,1)\times(0,1)$. Let $u\in L_2(\Omega)$ defined by $u(x,y) = 1$ for $x>y, u(x,y) = 0$ for $x\leq y$ Is there a way we can conclude ...
1
vote
1answer
71 views

Show $A$ is self-adjoint and $ f= Au$ in weak sense.

Hoi, consider $L_2(\Omega)$ with $\Omega = (0,1)\times (0,1)$ and let $u\in L_2(\Omega)$ be defined as $u(x,y) = 1$ for $x>y$ and $u(x,y) = 0$ for $x\leq y$. Let $A = \partial_x^2 - \partial^2_y$ ...
0
votes
1answer
35 views

Identifying 2 spaces of distributions

Hoi, I want to show that the space $C^{\infty}(\Omega)'$ of continuous linear functionals on $C^{\infty}(\Omega)$ can be identified with the subspace $\mathcal{E}'(\Omega)$ (distributions with ...
1
vote
1answer
47 views

Inequalities for point distribution

Let $u$ be a distribution on $\mathbb{R}^n$ with support = $\left\{0\right\}$. Then there exists $N$ such that $u$ has order $N$. Let $\chi\in C_0^{\infty}(\mathbb{R}^n)$ a smooth function with ...
2
votes
2answers
130 views

distribution with point support

Let $u$ be a distribution on $\mathbb{R}^n$ with support = $\left\{0\right\}$. Then there exists $N$ such that $u$ has order $N$. Let $\chi\in C_0^{\infty}(\mathbb{R}^n)$ a smooth function with ...
2
votes
1answer
110 views

Fourier transform of locally integrable function

I have a question and I don't know if it is true. Is any locally integrable function a sum of an $L_{1}$ function and another nice function (perhaps, an $L_{2}$ function). This is related to the ...
3
votes
1answer
108 views

Distributional limit

Let $u_t(x) = t^Ne^{itx}$ for $x\geqslant 0$ and $u_t(x)=0$ elsewhere. I want to calculate the distributional limit $\lim_{t\to\infty}u_t(x)$. How would one approach such problem. I just started a ...
5
votes
1answer
1k views

what do test function mean?

I am trying to learn weak derivatives. In that we call $\mathbb{C}^{\infty}_{c}$ function as test function and we use this function in weak derivatives. I want to understand why these are called test ...
3
votes
1answer
372 views

Prove that $f$ does not have a weak derivative

Consider a function $f:\mathbb{R} \rightarrow [0,1 ]$ defined by: $\begin{equation*} f(x)=\left\{ \begin{array}{rl}0 & \text{if } x\leq 0,\\ 1 & \text{if } x\geq 1, \\ 1/2 & \text{if } ...