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).

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2
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43 views

Fourier transform and non-standard calculus

The Fourier transform is not as easily formalizable as the Fourier series. For example, one needs to introduce tempered distributions to define the Dirac delta-function. Also, it is impossible to view ...
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1answer
20 views

Product of $C^\infty$ and $\mathcal D$

On my book there is the following statement: We can define the product of a distribution $u \in \mathcal D'(\Omega)$ and a function $\psi \in C^\infty(\Omega)$ in the following way: ...
1
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1answer
39 views

Distribution, equation

I need to find all solutions in $\mathcal {D'}(\mathbb{R})$( distributions $T$ ) of this equation: $id \cdot T' = 0$ $T'(\varphi ) = - T(\varphi ')$ I've already solved a similar equation $id ...
0
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1answer
44 views

Find distributions whose second derivative is Dirac delta

How can I find distributions $T \in \mathcal{D'}(\mathbb{R})$ such that $T'' = \delta _0$ ? I know that $D^2T(\phi) = T(D^2 \phi)$. $\phi \in C^{\infty}(\Omega, \mathbb{R}), supp \ \phi$ is compact ...
0
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2answers
40 views

distributional derivative of $x^{-1/2}$.

Consider the distribution defined for $\phi \in C_{c}(\mathbb{R})$ by $$T(\phi) = \int_{-\infty}^{\infty} |x|^{-1/2} \phi(x) dx.$$ Compute its derivative $T^{\prime}(\phi)$. Attempt: I use ...
2
votes
1answer
109 views

For what parameters does a sequence converge in $S$

Let $S$ be space of rapidly decreasing functions $f\in C_0^\infty(\mathbb R^n)$, that for any multi-indices $\alpha$ and $\beta$ there is a constant $M_{\alpha,\beta}$ such that $$|x^\alpha D^\beta ...
3
votes
1answer
83 views

does the function $|\sin(x) |$ define a tempered distribution? if so compute the fourier transform

I need to check if the function $|\sin(x)|$ defines a tempered distribution and find the fourier transform of the distribution. I think it defines it because it is summable on every compact ...
1
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1answer
40 views

Distribution as limit of quotient

Let $u \in \mathcal{D}'(\mathbb{R})$. Show that: $$\frac{u-\tau_{x}u}{x} \to \mathcal{D}u$$ in $\mathcal{D}'(\mathbb{R})$ when $x \to 0$. Here, we define $\tau_{s}f(x)=f(x+s)$. It's an exercise ...
0
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2answers
41 views

Distributional derivative of $\sin { (\pi|x|/2))} \chi_{\{|x|<2\}}$

I have to find the first and the second distributional derivative of the function : $$f(x) = \begin{cases} \sin { (\pi|x|/2))} \quad & |x| <2 \\ 0 \quad & |x| \geq 2 \end{cases}$$ but ...
0
votes
3answers
82 views

Distributions defined by $C_0^\infty(\mathbb{R})$ enough to distinguish $f_1,f_2\in L^1(\mathbb{R})$?

Let $f_1,f_2$ be Lebesgue-summable functions on the real line. I was wondering whether space $C_0^\infty(\mathbb{R})$ of infinitely differentiable compactly supported functions, intended as ...
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0answers
34 views

How should I use Sobolev embedding to prove Schwartz representation theorem?

From Richard Melrose, From Microlocal analysis to global analysis, Chapter 1. Show that, for any $p\in \mathbb{R}$ the map $$ R_{p}:S(\mathbb{R}^{n})\ni \phi\rightarrow ...
3
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0answers
32 views

What is the topological interpretation of continuity of distributions?

I was given this definition of continuity in the distributional sense. A distribution $T$ over the space of test functions $\mathcal{D}$ is continuous if for every sequence of test functions $\{ ...
4
votes
1answer
91 views

Can Fourier transform be seen as a decomposition over a basis in a space of tempered distributions

Fourier series of a function that belongs to $L^2([0,T])$ can be seen as a decomposition of this function over an (orthonormal) basis in the Hilbert space $L^2([0,T])$. Fourier transform of a ...
10
votes
2answers
280 views

Is this sequence bounded ? (An open problem between my schoolmates !)

Let $f$ be a smooth function (say $\mathcal{C}^{\infty}$) in its two real variables ($t$ and $T$). I consider the following sequence defined by $$A_n:=\lim_{T \to \infty} \int_{0}^{1} e^{-n t} ...
3
votes
0answers
62 views

What is a good way to prove the weak form of the Sobolev embedding theorem?

From Richard Melrose, From microlocal to global analysis, Chapter 1, Problem 11. Suppose $u\in L^{2}(\mathbb{R}^{n})$ and that $$ D_{1}D_{2}\cdots D_{n}\mu\in (1+|x|)^{-n-1}L^{2}(\mathbb{R}^{n}) $$ ...
0
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1answer
39 views

Proof on delta sequences

In a coursetext of mine there is a theorem that says: when $f(x)$ is locally integrable ($f(x)\in\text{L}^1_{\text{loc}}(\mathbb{R}^d$), $f(x)\geq0 \,\forall x$ and $\int f(x) dx=1$ then it follows ...
0
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0answers
28 views

Operator of Taylor series as a distribution?

I would like to prove a statement: $$T:=\sum_{k=0}^\infty a_k\partial_x^k\delta_0\not\in\mathscr{S}'(\mathbb{R}^n)$$ and, in contrast, $$T_n=\sum_{k=0}^n ...
3
votes
2answers
53 views

Correct definition of convolution of distributions?

Wikipedia states, that the definition of convolution of function $f$ with a distribution $T$ is $$\langle T\ast f,\varphi\rangle=\langle T,\tilde{f}\ast\varphi\rangle$$ where $\langle ...
1
vote
0answers
43 views

Represent Dirac distruibution as a combination of derivatives of continuous functions?

It seems from Rudin's functional analysis (P168, Thm6.27), the Dirac distribution $\delta_0$ on $R^1$ can be written as $$ \delta_0=f_0+f'_1+f''_2, $$ for $\{f_i\}_{i=0}^2\in C(R^1)$, there $'$ ...
1
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0answers
18 views

exactly represent Dirac distribution as derivatives of continuous function [duplicate]

It seems that in Rudin's functional analysis (P168, Thm6.27) that the Dirac distribution $\delta_0$ on $R^1$ can write as $$ \delta_0=f_0+f'_1+f''_2, $$ for $\{f_i\}_{i=0}^2\in C(R^1)$, there $'$ ...
1
vote
2answers
66 views

Is dirac comb a tempered distribution?

Prove or disprove the statement $$ III:=\sum_{k=-\infty}^\infty\delta_k\in \mathscr{S}'(\mathbb{R}^n)$$ where $\delta_k\varphi:=\varphi(k),\,\,\forall\varphi\in\mathscr{S}(\mathbb{R}^n)$ I tried to ...
1
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1answer
37 views

Distributional logarithm

I am reading Distribution Theory courently but there is part that I can't pass: Last equation in $(1.204)$ makes no sence for me becouse as far as I know: ...
3
votes
1answer
97 views

How to show $e^{x}\cos[e^x]$ is a tempered distribution?

From Melrose, Lecture notes on Microlocal Analysis, Chapter 1. I was asked to show that the function $$ u(x)=e^{x}\cos[e^{x}] $$ is a tempered distribution. I tried to use the definition that there ...
0
votes
1answer
111 views

Convolution with dirac delta - proof

I have dirac delta defined as $\delta(f)=f(0)$, where $f(x)$ is an arbitrary function. I have defined convolution of distribution and function as $T\ast f=T(\tilde{f}\ast\varphi)$, where ...
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0answers
74 views

Characterization of $H^{-1}(\mathbb{R}^N)$ by Fourier transform

We denote by $H^{-1}(\mathbb{R}^N)$ the dual space to $H_0^1(\mathbb{R^N})$. Let $$\langle f,v \rangle = \int_{\mathbb{R}^N} fv \hspace{1cm} (f,v \in L^2(\mathbb{R}^N)).$$ It is known that if $f \in ...
0
votes
1answer
30 views

Dirichlet problem, solvability

Let $\Omega\subset \mathbb R^n$ and $f\in L^2(\Omega)$. We consider $\{u\in C^1(G):||u||_{1,2,G}<\infty \}$,where $||u||_{1,2,G}$ is the "sobolev-norm" with parameter $1,2$ on $G$. Then $H^1(G)$ ...
2
votes
1answer
52 views

How to evaluate a double integral with two Dirac functions?

Here I have a problem, is the solution the same if I integrate every one? part by part? $$\int_0^Te^{-(s+\mu\lambda^2 ) t} \int_0^l\left[\delta(x-R)\delta(t-tj)\varphi(x) \, dx\, dt\right]$$ I've ...
0
votes
1answer
24 views

Show that a particular sum converges and defines a distribution (example from Friedlander)

For $\phi \in C^\infty_0(\mathbb{R})$, define the quantity $\langle u, \phi \rangle$ by: $$ \langle u, \phi \rangle = \lim_{m \to \infty} \left[ \left(\sum_{k =1 }^m \phi\left(\frac{1}{k}\right) ...
1
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2answers
40 views

Convergence in $D'$

I was given by my professor of mathematical methods for physicist, a notion of convergence in $D'(\Omega)$, the space of distributions on $D(\Omega)$. Namely: A sequence of distributions $T_n\in\ ...
0
votes
0answers
27 views

Mistake in reasoning: $Pv\frac{1}{x} = 0$??

I'm making some mistake in my reasoning, which leads to $Pv\frac{1}{x} = 0$ (in distributional sense): $$<Pv\frac{1}{x},\phi> = lim_{\epsilon \rightarrow ...
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0answers
15 views

Bump function construction with positive Fourier transform [duplicate]

Fellow math people, I am looking to construct a bump function with a positive and rapidly decaying Fourier transform. In particular, the function f should satisfy: (1) f non-negative and smooth and ...
2
votes
1answer
42 views

Verifying Distribution Equivalence for Fourier Series Expansion

In my lecture notes, given a periodic distribution $T \in (C_{per}^\infty([-\pi,\pi]^n))'$, the Fourier coefficients are defined by $$\hat T(m) = T({1 \over (2\pi)^n}e^{-i m \cdot x}),$$ for $m \in ...
0
votes
2answers
136 views

In what sense is $\int_{-\infty }^{\infty } \frac{x}{x^2+1} \, dx = \pi i$?

Suppose we want to give a meaning to the divergent integral $$I = \int_{-\infty }^{\infty } \frac{x}{x^2+1} \, dx,$$ perhaps in the sense of distributions or something (similarly to how $\int_{-\infty ...
1
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2answers
30 views

Support of polynomial distributions

Assume $u\in\mathcal{S}'(\mathbb{R}^n)$ is a tempered distribution such that $\widehat{u}$ is compactly supported and $u^k$ defines a distribution for each $k=1,\cdots,m$. Let $p_1,\cdots,p_m$ be ...
3
votes
2answers
159 views

Topology on the space of test functions

I try to read into the theory of distributions and there is one thing which bothers me. I read that a distribution is a linear, continuous functional from the space of test functions, which, depending ...
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0answers
11 views

Generalised function equality

Can you explain in which sense, for a generalized function $$ Y^+(\phi):=\int_0^\infty e^{i\phi t}dt$$ we have $(-i\phi)Y^+(\phi)=1$ ? Here $\phi \in S^1$. I clearly get that $\int_0^\infty e^{i\phi ...
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0answers
29 views

How to create a linear set of proportions summing to 1?

I'm not even really sure what this concept is called. But my objective is to create, for any n, a linear distribution of numbers less than 1 summing up to and plateauing at 1. It would have to work ...
0
votes
1answer
24 views

Support of tempered distribution under exponetiation and differentiation

Suppose $u$ is a tempered distibution in $\mathbb{R}^n$. How are supp$(\widehat{u})$ and support of $\sum_{|\alpha|\leq k}\frac{\partial^{\alpha}\widehat{u^n}}{\partial x^{\alpha}}$ compared , where ...
0
votes
1answer
33 views

Support of polynomial distribution

Let $P(x_1,\cdots,x_n)$ be a polynomial in $\mathbb{R}^n.$ What is supp$(\widehat{P})$ when $P$ viewed as a tempered distribution. Can supp$(\widehat{P})$ be the boundary of an sphere?
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22 views

$L^{2}$ convergence and converence of distribution

Suppose that $f_{n}(x)$ are a sequence of $L^{2}$ functions which converge to a function $f(x)$ in the $L^{2}$ sense. Show that it also converges weakly in the sense of distributions, ie for any test ...
2
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1answer
59 views

Charge distribution on an arbitrarily shaped conductor

From physics we know that given a charged conductor put in vacuum (no external electric fields), the charge distribution on its surface is approximately proportional to the curvature of the surface on ...
0
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1answer
24 views

convergences in $\mathcal {S'}$

strong textLet $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with $ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: |\xi|\leq 2, \phi(\xi)=1$ if $|\xi|\leq 1.$ My Question is: ...
0
votes
0answers
24 views

Fourier transform of $\frac{1}{x_1^2+x_2^2+x_3^2}$ [duplicate]

How can I find Fourier transform of $$\frac{1}{x_1^2+x_2^2+x_3^2}?$$
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0answers
117 views

A generalization of the Sokhotski–Plemelj theorem

The Sokhotski–Plemelj theorem on the real axis states: $$ \lim_{\varepsilon\rightarrow 0^+} \int_a^b \frac{f(x)}{x\pm i \varepsilon}\,dx = \mp i \pi f(0) + \mathcal{P}\int_a^b \frac{f(x)}{x}\, dx $$ ...
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0answers
45 views

Dirac delta composed with a function and implicit equation for the roots

I'm considering an expression of the form $$\int_{-\infty}^\infty dx G(x) \delta(x^2-f(x)^2) $$ where $G$ and $f$ are two unrelated smooth functions of $x$. Now I know that when $f$ is a positive ...
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0answers
28 views

How to decompose tempered distribution by entire analytic functions?

Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with $ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: |\xi|\leq 2, \phi(\xi)=1$ if $|\xi|\leq 1.$ Let $j\in \mathbb N$ and ...
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0answers
202 views

computing how distributional derivatives behave under coordinate transformations

Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ with smooth ($C^\infty$) boundary. For $k > 1$, assume that $u \in C^{k-1}(\Omega)$ such that its order $k$ derivatives exist in $\Omega$ and ...
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2answers
44 views

Absolutly integrable functions are injective to tempered distribution?

We had a theorem that$$\mathcal{L}^1(\mathbb{R}^n)\hookrightarrow \mathscr{S}'(\mathbb{R}^n)$$ Where $\mathscr{S}'$ is the space of tempered distributions. In the proof our lecturer constructed a ...
0
votes
1answer
71 views

Finding Distributional Solution

In the range $0 \leq r < \infty$, find the solution of the equation $$\frac{d^{2}u}{dr^{2}} + \frac{2}{r} \frac{du}{dr} - \frac{n(n+1)}{r^{2}} u = a \delta(r-R),$$ where $n$ is an integer and ...
0
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0answers
58 views

symmetric and anti symmeric distribution - sqrt function on it

I've got question for homework and I'm not sure about it, I appreciate your help. 1. assuming distribution is anti symmetric, if we apply the function sqrt on it, will we get symmetric distribution ...