Tagged Questions
5
votes
1answer
54 views
Dirac Delta or Dirac delta funtion?
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
25 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 ...
0
votes
0answers
21 views
Show that$ a$ is a differential of order $m$.
Lat $a = a(x,\zeta) \in S_{1,0}^m(\mathbb{R}^n,\mathbb{R}^n)$. Write $n=n_1+n_2$ with $n_2\geq 1$ and $\zeta = (\zeta_1,\zeta_2)$ with $\zeta_i\in \mathbb{R}^{n_i}$. Suppose that $a$ does not depend ...
1
vote
1answer
33 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
$$
|D^\alpha f(\xi)| \leq ...
0
votes
1answer
34 views
A clarification about BV functions.
From definition, a locally integrable $ u \in BV(\Omega) $ if its distribution derivative is given by a signed Radon measure. That is there exists $ \mu $ such that for any $ \phi \in ...
2
votes
2answers
80 views
Show existence of a continuous $k$ on $\mathbb{R}^2$ such that $(u,\phi)= \int_{\mathbb{R}^2}k\phi dx$ for all $\phi$
(b). Let $u$ be a distribution on $\mathbb{R}^2$. Assume there exists a continuous function $h$ on $\mathbb{R}^2$ such that $(u,\Delta \phi) = \int_{\mathbb{R}^2}h\phi dx $ for all $\phi\in ...
2
votes
1answer
49 views
Showing that $H_{\text{loc}}^2(\mathbb{R}^2) \subset C^0(\mathbb{R}^2)$
I want to show that $H_{\text{loc}}^2(\mathbb{R}^2) \subset C^0(\mathbb{R}^2)$
$C^0$ is the space of continuous functions, and $H_{\text{loc}}^2(\mathbb{R}^2)$ the set of distributions $u\in ...
3
votes
0answers
58 views
Computing the Fourier transform (in the sense of tempered distributions) from a sequence of samples
Let $f\in L_{loc}^2 (\mathbb{R})$ be a function of period $2\pi$. I'd like to get some help proving the following identity:
$\hat{f}=\Sigma_{n=-\infty}^{\infty} \hat{f}(n)\delta_n$
With some ...
1
vote
2answers
61 views
Show existence of continuous functions $f$ with $f''=\delta_0-\delta_1$
Let $u$ be the distribution on $\mathbb{R}$ given by $$u=\delta_0-\delta_1 $$
(a) show there exists a continuous function $f$ such that $f''=u$ and indicate such one. I thought of doing this with ...
0
votes
1answer
34 views
Operation on distributions
I'm currently studying a course on Advanced Real Analysis for a master degree, and our professor handed to everyone of us a 40-page book. I'm major in Algebra, so I'm not really comfortable with this ...
2
votes
0answers
56 views
Inversion formula for Schwartz-space $\mathcal{S}$.
Let $T\phi = \overline{\mathcal{F}}\mathcal{F}\phi$ for $\phi \in \mathcal{S}(\mathbb{R}^n)$ (rapidly decreasing functions or Schwartz-functions, $\mathcal{F}$ fourier transform and ...
0
votes
1answer
108 views
Is it a dirac-delta?
Hoi, consider $\displaystyle u= \frac{1}{|x|}e^{-|x|}$ for $x\in \mathbb{R}^3$, then one can see
that $\Delta u = u$ for $|x|>0$ ( which one can see by transferring $u$ to spherical coordinates).
...
1
vote
1answer
60 views
Inverse fourier transform 3 dimensions
Hoi, I want to show that for $n=3$ that $$\mathcal{F}^{-1}\left(\frac{1}{1+|s|^2}\right) = \frac{1}{4\pi |x|}e^{-|x|} $$
As a hint I've been given: Its the unique solution to the equation ...
8
votes
1answer
71 views
Various kinds of derivatives
Let $f\colon \mathbb{R}\to \mathbb{R}$ be a measurable function. Let us introduce the following notions of "derivative" of $f$.
Classical derivative. The unique function $f'_c$ defined pointwise by ...
1
vote
1answer
60 views
Calculation of the Laplacian of a function in $\mathbb{R}^3$.
I have to calculate the Laplacian distributional sense) of the following function
$$\frac{e^{i\sqrt{\lambda}|x|}-1}{4\pi|x|}$$
with $\lambda>0$, $x\in\mathbb{R}^3$. I've procedded in the following ...
0
votes
1answer
30 views
Showing $H^1(I) \subset C^{1/2}(\overline{I}) $
Hoi, let $H_1(I)$ the sobolev space on the interval $I = (\alpha,\beta)\subset \mathbb{R}$
http://en.wikipedia.org/wiki/Sobolev_space See here for more on sobolev spaces.
$H_m(I)$ contains all ...
0
votes
1answer
30 views
Boundedness of operator
I want to show that the following Linear operator $L$ is bounded and surjective:
$L: H^2(I) \to \mathbb{C}^4$ where $I = [\alpha,\beta]\subset \mathbb{R}$
given by $L(u) = ...
0
votes
1answer
84 views
Is this integral 0?
Let $\phi \in C_0^{\infty}(\Omega)$ with $\Omega = (0,1)\times(0,1)$. Let $u\in L_2(\Omega)$ defined by $u(x,y) = 1$ for $x>y, u(x,y) = 0$ for $x\leq y$
Is there a way we can conclude ...
1
vote
1answer
58 views
Show $A$ is self-adjoint and $ f= Au$ in weak sense.
Hoi, consider $L_2(\Omega)$ with $\Omega = (0,1)\times (0,1)$ and let $u\in L_2(\Omega)$ be defined as
$u(x,y) = 1$ for $x>y$ and $u(x,y) = 0$ for $x\leq y$. Let $A = \partial_x^2 - \partial^2_y$
...
0
votes
1answer
32 views
Identifying 2 spaces of distributions
Hoi,
I want to show that the space $C^{\infty}(\Omega)'$ of continuous linear functionals on $C^{\infty}(\Omega)$ can be identified with the subspace $\mathcal{E}'(\Omega)$ (distributions with ...
1
vote
1answer
37 views
Inequalities for point distribution
Let $u$ be a distribution on $\mathbb{R}^n$ with support = $\left\{0\right\}$. Then there exists $N$ such that $u$ has order $N$. Let $\chi\in C_0^{\infty}(\mathbb{R}^n)$ a smooth function with
...
2
votes
2answers
58 views
distribution with point support
Let $u$ be a distribution on $\mathbb{R}^n$ with support = $\left\{0\right\}$. Then there exists $N$ such that $u$ has order $N$. Let $\chi\in C_0^{\infty}(\mathbb{R}^n)$ a smooth function with
...
0
votes
0answers
35 views
Computing a distribution at a point (Heat Equation).
I’d like some help to find the value of
$$
\langle (\partial_{t} - \Delta) G^{\epsilon},\phi \rangle,
$$
where
$ {G^{\epsilon}}(x,t) := G(x,t) \cdot {\chi_{(\epsilon,\infty)}}(t) $,
$ G(x,t) := (4 ...
2
votes
1answer
45 views
Fourier transform of locally integrable function
I have a question and I don't know if it is true. Is any locally integrable function a sum of an $L_{1}$ function and another nice function (perhaps, an $L_{2}$ function). This is related to the ...
3
votes
1answer
51 views
Distributional limit
Let $u_t(x) = t^Ne^{itx}$ for $x\geqslant 0$ and $u_t(x)=0$ elsewhere. I want to calculate the distributional limit $\lim_{t\to\infty}u_t(x)$.
How would one approach such problem. I just started a ...
4
votes
1answer
101 views
what do test function mean?
I am trying to learn weak derivatives. In that we call $\mathbb{C}^{\infty}_{c}$ function as test function and we use this function in weak derivatives. I want to understand why these are called test ...
3
votes
1answer
113 views
Prove that $f$ does not have a weak derivative
Consider a function $f:\mathbb{R} \rightarrow [0,1 ]$ defined by:
$\begin{equation*}
f(x)=\left\{
\begin{array}{rl}0 & \text{if } x\leq 0,\\
1 & \text{if } x\geq 1, \\ 1/2 & \text{if } ...
0
votes
1answer
118 views
Limit of distributions of principal value
What is the limit in $D'(\mathbb{R})$ (i.e. in the distribution sense) of
\begin{equation}
\lim_{t \rightarrow +\infty}\frac{e^{ixt}}{x+i0}
\end{equation}
where $x+i0=p.v.(\frac{1}{x})-i\pi\delta(x)$ ...
2
votes
1answer
126 views
About a property of the Dirac delta function
How can I show that there is no $u$ satisfying both (i) and (ii):$$(i) \; u \in L^p (\Bbb R^n )$$ and $$(ii) \int_{\Bbb R^n} \delta (x) \phi(x) dx= \int_{\Bbb R^n} u (x) \phi(x)dx\; ( \forall \phi ...
1
vote
1answer
142 views
Inverse function with Dirac Delta
We know that the inverse function of
$ y=\log(x) $ is $ y =\exp(x) $.
However, what would be the inverse of
$ y=\log(x)+ \sum_{n=1}^{\infty}\delta (x-n) $?
I have tried with Mathematica, and ...
3
votes
1answer
66 views
Prove the Borel Lemma
I'm trying to prove the Borel Lemma, which is:
For every series $a_0,a_1,a_2,\dots$ in $\mathbb{C}$ exists $f \in C^{\infty}(\mathbb{R})$ such as $$ f^{(k)}(0) = a_k $$ for every $k \in ...
0
votes
1answer
93 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
397 views
Distributions supported on a single point
Let $d=1$.
(i) Show that if $\lambda$ is a distribution and $n\geq1$ is an integer, then $\lambda x^n=0$ if and only if $\lambda$ is a linear combination of $\delta:=\delta_{\{0\}}$ and its first ...
3
votes
1answer
109 views
Show the usual Schwartz semi-norm is a norm on the Schwartz space
Let $f \in C^\infty(\mathbb R)$. Define the semi-norm
$$
\|f\|_{a,b}=\sup_{x \in \mathbb R} |x^af^{(b)}(x)|
$$
where $a,b \in \mathbb Z_+$, and $f^{(b)}$ is the $b$-th derivative of $f$.
Show ...
2
votes
1answer
125 views
Schwartz space: semi norm estimate on translation
the following family of semi norms is commonly used to introduce the space of Schwartz functions $\mathcal{S}(\mathbb{R}^n)$:
$$
\|\phi\|_N := \sup_{\substack{x \in \mathbb{R}^n \\ |\alpha|\,,|\beta| ...
1
vote
1answer
61 views
Proving a sequence of distributions converge in $C^{-\infty}(\mathbb{R})$
Question: I have a sequence $(T_n)$, where $T_n$ is given by the locally integrable function $ne^{-\dfrac{n^{2}x^{2}}{2}}$, converges in $C^{-\infty}(\mathbb{R})$ and compute its limit.
I suspect ...
2
votes
3answers
189 views
What is the appropriate topology on $C_c^\infty (\mathbb{R}^d)$?
Let $\{ U_k:k\in \mathbb{N}\}$ be an increasing sequence of open subsets of $\mathbb{R}^d$ whose union is $\mathbb{R}^d$ and such that each $K_k:=\overline{U_k}$ is compact and $K_k\subseteq U_{k+1}$.
...
0
votes
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}$?
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 ...
3
votes
2answers
185 views
A sufficient condition for a function to be of class $C^2$ in the weak sense.
Let $f\colon\mathbb{R}\to\mathbb{R}$ be a continuous function with weak derivative (i.e. the derivative in the sense of distribution) in $C^1(\mathbb{R})$. Does this condition imply that $f$ is two ...
1
vote
1answer
125 views
Question about support of distributions
I'm reading "Pseudo-differential Operators and the Nash-Moser Theorem" and at the top of on page 8 they write: "Finally, we note that if $u \in C^0(\Omega)$, then the support of $u$ defined above ...
1
vote
1answer
152 views
Delta function representation
Suppose I have a set of functions $(f_\epsilon)$ such that as $\epsilon\to 0$, $f_\epsilon\to F$ s.t.
$F(x)=0$ for $x\neq 0$ and $F(x)=\infty$ for $x=0$;
$\int_{-\infty}^\infty f_\epsilon(x) dx=1$ ...
6
votes
1answer
189 views
Questions about Fubini's theorem
I learned the following from Hunter's Applied Analysis.
Denote the Schwartz space
$${\mathcal S}({\mathbb R}^n):=\{\varphi\in C^{\infty}({\mathbb R}^n):\sup_{x\in{\mathbb ...
8
votes
2answers
542 views
How to prove that the Cantor ternary function is not weakly differentiable?
I am using the standard cantor ternary function $f$ here, as cited in this Wikipedia page.
It is an example of continuous, monotone increasing, but not strictly monotone increasing function with zero ...
8
votes
2answers
182 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 ...
