Use this tag for questions about distributions (or generalized 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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4
votes
1answer
141 views

A problem from distribution theory.

Let $f$, $g\in C(\Omega)$, and suppose that $f \neq g$ in $C(\Omega)$. How can we prove that $f \neq g$ as distributions? Here's the idea of my proof. $f$ and $g$ are continuous functions, so they ...
3
votes
1answer
97 views

prove $ \mathcal F(f) = c_1\delta + c_2 \delta' + c_3\delta'' + T_g $ with $f(x)=|x^2 -1|$

let $f(x)=|x^2 -1|$ be a tempered distribution (i proved it) , and calculated its 3rd derivation (as a distribution) and then this stopped me : prove that we have : $$ \mathcal F(f) = ...
2
votes
1answer
89 views

prove the existence of $c\in \mathbb C$ such as $\mathcal F(g)=c\delta + {1\over2i\pi}V_p({\mathcal F(f)\over x}) $

prove that it exist $c\in \mathbb C$ such as $\mathcal F(g)=c\delta + {1\over2i\pi}V_p({\mathcal F(f)\over x}) $ with $\mathcal F$ fourier transform , $f\in \mathbb D(\mathbb R)$ , $\delta$ ...
1
vote
1answer
186 views

distribution - exercises

Let $f \in \mathcal{C}(\mathbb{R}) \cap L^1(\mathbb{R}).$ We suppose that $f$ is bounded and even $(f(-x)=f(x))$. let $\lambda \in \mathbb{R}.$ We assume that: $\int_{-1}^1 \dfrac{1 - f(\lambda ...
1
vote
1answer
408 views

Integral with delta Dirac power

Is it possible to calculate the integral: $$J=\int_{-\infty}^{+\infty}f(x)\delta(x-x_0)^kdx$$ wih $k\in\mathbb{R}$? I know that in the Colombeau algebra the distribution $\delta(x)^2$ is defined. What ...
4
votes
2answers
391 views

Definition of convergence in $C^\infty(\Omega)$

I am not convinced or may be don't understand, the way they define convergence and then topology as a consequence of convergence. $\Omega$ is open subset of $\Bbb R^n.$Define standard topology on ...
4
votes
2answers
5k views

Dirac delta in polar coordinates

Given $$x=r\,\cos\theta\\y=r\,\sin\theta$$ and $$x'=r'\,\cos\theta'\\y'=r'\,\sin\theta'$$ how can I express $$\delta(x'-x)\delta(y'-y)$$ in terms of the polar coordinates? And the more general ...
2
votes
1answer
89 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 $n$ ...
0
votes
1answer
96 views

Does convergence in $C(\Bbb R)$ imply convergence in $\mathcal D'(\Bbb R)$

Sequence of continuous functions $(f_k)$ which converge to $f$ in $C(\Bbb R)$. Does it imply that $f_k$ will converge to $f$ in $\mathcal D'(\Bbb R)$ as well? Or we need something more to make this ...
3
votes
1answer
261 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 ...
11
votes
3answers
925 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.
2
votes
1answer
140 views

Regularizing a solenoidal vector field $u\in L^p(\Omega)^N$.

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain and suppose that $u\in L^p(\Omega)^N$, $p\in (1,\infty)$. Assume that in the sense of distributions, $\operatorname{div}u=0$ where ...
2
votes
4answers
151 views

How can you show that $\delta′=f(0)\delta′−f′(0)\delta$ for a function f that is infinitely differentiable?

Assume that $f$ is infinitely differentiable. Let $\delta$ be the (Dirac) delta functional. I know that $f\delta = f(0)\delta$, but I'm not sure how to derive the equation ...
1
vote
2answers
50 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
58 views

Distribution $P_{\frac{1}{x^3}}$

How to show that for $\phi\in D(R)$, $<P_{\frac{1}{x^3}},\phi>=v.p.\int_{-\infty}^{\infty}\frac{\phi(x)-x\phi'(0)}{x^3}$ defines a distribution? It is easy to check that $P_{\frac{1}{x^3}}$ ...
7
votes
2answers
582 views

How do we define the $L^p$ norm of a tempered distribution?

I am finishing up a class on function theory and I am trying to reconcile a few statements that I have seen. Let us define $L^p(\mathbb R^n)$ to be the set of measurable functions $f$ so that ...
0
votes
1answer
112 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 ...
2
votes
1answer
49 views

Question about distributions

How to show that function $\phi\in D'(R)$ can be represented as derivative of function $\phi_1\in D(R)$ if and only if: $\int_{-\infty}^{\infty}\phi(x)dx=0$. One direction is easy: if $\phi=\phi'_1$, ...
2
votes
1answer
185 views

Finding distributional limit

How to find $\lim_{\varepsilon\rightarrow 0+}f_{\varepsilon}$ in $D'(R)$, if $f_\varepsilon$ is defined as: $f_\varepsilon(x)=\frac{1}{\varepsilon^3}$ for ...
4
votes
1answer
164 views

When can you integrate a derivative?

Let us say I have an expression $$\frac{d}{dt}f = g$$ where the derivative is taken in a weak sense (of distributions). Can I integrate this from $0$ to $t_0$ and get $$f(t_0) - f(0) = ...
1
vote
1answer
69 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
95 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
112 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
80 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 ...
2
votes
1answer
61 views

Asymptotic behaviour of solutions to elliptic PDE

Let $u$ be a solution (in the distributional sense) of $$ \Delta u = \delta_r $$ on $\Omega \subset \mathbb{R}^2$ open, $r \in \Omega$. Let $w$ be a solution of $$ Aw = \delta_r $$ where $A = ...
1
vote
0answers
40 views

Solve $P(\frac{d}{dx})u=f$ when $f$ is a distribution with compact support

I need some help for the following statement Let $P$ be a polynomial, and $P(\frac{d}{dx})u=f$, where $f$ is a distribution with compact support. Then it has a distributional solution $u$ with ...
5
votes
1answer
283 views

Taylor series and tempered distributions

Suppose we have a function $\psi$ in $\mathbb{R}$ other than a polynomial which is equal to its Taylor Series for every point in $\mathbb{R}$. Why is the following statement valid? When we interpret ...
2
votes
2answers
79 views

Finding limit in $D'(R)$

Find a limit when $\varepsilon\rightarrow +0$ of the following distribution: $$f_\epsilon=\frac{\varepsilon x}{(x^2+\varepsilon^2)^2}$$ I tried to solve this by putting $x=\varepsilon t$, but ...
3
votes
1answer
162 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
92 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 ...
2
votes
1answer
87 views

$\phi$ has compact support in $\mathbb{R}^n$ does not imply $\phi (\xi + \eta)$ has compact support in $\mathbb{R}^n\times\mathbb{R}^n$

Let $\phi$ be a $C^\infty$ function with compact support in $\mathbb{R}^n$. Some introductory books on distribution theory I'm reading say that the function $(\xi,\eta)\mapsto \phi (\xi + \eta)$ not ...
2
votes
1answer
151 views

convolution-distributions

We denote by $E'(\mathbb{R})$ the set of distribution with compact support , and $\mathcal{D}(\mathbb{R})$ is the set of function $\mathcal{C}^{\infty}$ with a compact support. 1) I want to compute ...
0
votes
1answer
122 views

convolution-distribution

i want to compute the product of convolution $1 * (\delta' * H)$ where $\delta$ is distribution of Dirac and $H$ is function of Heaviside. first, we compute $\delta' * H.$ We have by definition that ...
2
votes
2answers
1k views

Sum over cosines = dirac delta - how to get the coefficients?

Given this formula: $$\sum\limits_{n=0}^\infty a_n \cos(n \pi x / d) = \delta(x-x_0)$$ Where $0 \leq x \leq d$. How can one calculate the coeffciients $a_n$? I googled and searched all kinds of ...
0
votes
1answer
67 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 ...
1
vote
1answer
234 views

Limit in a sense of distributions

How to find $\lim_{a\rightarrow\infty} f_a$ in $D'(R)$, for $a>0$, where $f_a:R\rightarrow R$ is defined by $f_a(x)=\begin{cases}\frac{\sin{ax}}{x}&x\neq 0 \\0&x=0\end{cases}$ Thanks in ...
1
vote
1answer
45 views

Question about convergence in $\mathcal D(\Bbb R)$

Let $\phi\in \mathcal D(\Bbb R)$. How to prove or disprove convergence of $\phi_n(x)=\frac{1}{n} \phi(nx)$ in $\mathcal D(\Bbb R)$? I tried to do this by definition (we have to check two conditions ...
3
votes
0answers
103 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 ...
2
votes
0answers
67 views

Distributions - please check my solution

I have to find a derivative in a distributional sense of the following function (known as Cantor's singular function) $$f(x)=\left\{ \begin{array}{l l l l l} 0, & \quad\text{$ x\leq 0 $}\\ 1, ...
8
votes
2answers
2k views

Laplacians and Dirac delta functions

It is often quoted in physics textbooks for finding the electric potential using Green's function that $$\nabla ^2 \left(\frac{1}{r}\right)=-4\pi\delta^3({\bf r}),$$ or more generally $$\nabla ...
0
votes
1answer
231 views

Show that $\delta(\xi-x)=\delta(x-\xi)$

How would you show $\delta(\xi-x)=\delta(x-\xi)$ if you know that $$\int _{-\infty}^{\infty}\delta(x)h(x)=h(0)$$ Also how would you then show more generally that if $f(\xi)$ is a monotonic ...
1
vote
1answer
88 views

Find $r(x)$ such that $r(x)L$ is self-adjoint

The differential operator $$L=a(x)\frac{d^2}{dx^2}+b(x)\frac{d}{dx}+c(x)$$ is not self adjoint. How would you find r(x) such that r(x)L is self adjoint. I know that this is self adjoint when $L=L^*$ ...
5
votes
4answers
419 views

delta functions $e^{x}\delta (x)=\delta (x)$

How would you prove that; $$e^{x} \delta (x)= \delta (x)$$ Is it anything to do with the following relationship; $$ \int_{-\infty}^{\infty} g'(x)h(x)\,dx = \int_{-\infty}^{\infty} g(x)h'(x)\,dx.$$ ...
0
votes
1answer
222 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
38 views

Even distribution?

What exactly does it mean for a distribution $u\in \mathcal{D}'(\mathbb{R})$ to be even? Does it mean that for even testfunctions $\varphi $ it holds that $\langle u, \varphi |_\mathbb{R_+} \rangle = ...
1
vote
1answer
526 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 ...
1
vote
2answers
84 views

Show that the order of $\delta'$ is one

I try to show that the order of $\delta'$ is one. Clearly $|(\delta', \phi)| = |\phi'(0)| \leq \sup \phi'(x)$ but if I got the definition of order right I have to show that $|\phi'(0)| \leq \sup ...
1
vote
1answer
168 views

Multiplication of distributions by smooth functions

Let $u\in D'(\mathbb{R})$ and $f\in C^{\infty}$. I'm trying to figure which of the following statements is true: I. If $f\restriction_{supp(u)}=1$ then $f\cdot u=u$. II. If ...
9
votes
2answers
203 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
139 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 ...