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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21 views

Integral involving rapidly decreasing functions [on hold]

Let $F,\varphi\in S(\mathbb{R}^n)$, the Schwartz space of rapidly decreasing functions. Is this enough to guarantee that the integral $$\int_{\mathbb{R}^n} F(x)\varphi(x)dx$$ is well-defined? Why or ...
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1answer
27 views

Intiution behind the derivative of dirac delta function

Let me first begin what I mean by saying the intuition behind the " $\delta'(x)$ ". For example the smooth approximations of the delta function looks like the following: (Left:the smooth ...
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1answer
27 views

Pre-dual of distributions with support in a closed subset

Usually, in order to define the space of distributions $\mathcal{D}'(\Omega)$ on an open subset $\Omega \subseteq \mathbb{R}^n$, one considers the space $\mathcal{D}(\Omega)$ of $C^\infty$-test ...
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1answer
72 views

How to prove that the Dirac delta is not a function?

I am currently taking a course on test functions and distributions and my task is to prove that the Dirac delta is not a function. Furthermore, I would also like to prove that it is continuous as a ...
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1answer
26 views

Defining a distribution

We fix the space $\mathcal{D}=\mathcal{C}^\infty_0(\mathbb{R}^n)$ as space of testfunctions. Let $(f_n)$ be a sequence of distributions with $\lim_{n\to\infty} f_n(\varphi)$ existing for all ...
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1answer
27 views

Differential Equation In Distributions

I need to find all $u \in \mathcal{D}'$ ( space of distributions) such that $ e^x (e^{-x} u ) ' = \delta_0 +1$. For any $\phi \in C_0^\infty(\mathbb{R})$ we have $\langle e^x (e^{-x} u ) ' , ...
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1answer
51 views

Fourier Transform of a Temperate Distribution

Let $f$ be a temperate distribution. Suppose that $f$ is a solution to the equation $ f'-f= \delta_0 +1 $. I want to find $ \hat{f}$... Here's what I did: Since $ f'-f= \delta_0 +1 $, then $ ...
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0answers
32 views

if $\Delta u = |\nabla u|^{3/2}$ then $u \in C^{\infty}$

Suppose $u \in H^1_{\text{loc}}(\mathbb{R^2})$. I want to show that if $\Delta u = |\nabla u|^{3/2}$ in the distributional sense then $u \in C^{\infty}$. I know that since $\nabla u \in L^2$ (I'll ...
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0answers
23 views

Distribution annihilated by a vector field

Let $u$ be a distribution in $\mathcal{D}'(M)$ (the continuous dual of $\mathcal{D}(M) = C_0^\infty(M ; \mathbb{C})$), where $M$ is a smooth manifold. Let also $X$ be a smooth vector field on $M$, ...
3
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0answers
65 views

Klein-Gordon field commutator integral?

Consider a Klein-Gordon field $\phi$, which satisfies $$(\Box+ \omega_0^2)\phi=0$$ on points $x \equiv \{x_0,\vec{x}\},y\equiv \{y_0,\vec{y} \}$ of 4D Minkowski-spacetime. The field commutator is $$ ...
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0answers
28 views

Why $\mathcal{D}(\Omega)$ be a topological vector space is important?

Let $\Omega\subset\mathbb{R}^N$ be an open set and $\mathcal{D}(\Omega)$ the set of all infinitely differentiable functions with compact support on $\Omega$. In the study of PDEs, we use the ...
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1answer
26 views

Show that a distributional solution of $\Delta u = f u $ is smooth for smooth $f$

As in the title - I would like to show that if $f$ is a smooth ($C^{\infty}$) function then for any distribution $u$ satisfying $$ \Delta u = fu$$ in the distributional sense we have, in fact, $u ...
2
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1answer
44 views

Delta distribution as a bounded linear functional

Let $\Omega \subset \mathbb{R}^n$ be a domain with $0 \in \Omega$. For which $n \in \mathbb{N}$ is the Dirac-delta distribution a bounded linear functional on $H_{0}^{1}(\Omega)$, that is to say, an ...
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1answer
24 views

Is the solution of a PDE always the convolution of the Green function?

If we are given Poisson's equation in a space: $$\nabla ^2 u=F$$ The solutions (those who admit Fourier transform) are given by: $$u(x)=\int_\mathbb{R^n} G(x,y)F(y)dy$$ Where $G$ is the Green ...
2
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0answers
34 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
17 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: ...
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1answer
27 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 ...
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1answer
29 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 ...
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2answers
28 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
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1answer
106 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
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1answer
47 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 ...
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1answer
37 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 ...
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2answers
28 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 ...
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3answers
80 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
20 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 ...
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0answers
29 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
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1answer
61 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 ...
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2answers
267 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} ...
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0answers
44 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 $$ D_{1}D_{2}\cdots D_{n}\mu\in (1+|x|)^{-n-1}L^{2}(\mathbb{R}^{n}) $$ ...
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1answer
26 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 ...
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0answers
27 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 ...
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2answers
39 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 ...
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0answers
30 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 $'$ ...
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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 $'$ ...
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2answers
53 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 ...
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1answer
34 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: ...
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1answer
52 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 ...
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1answer
52 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
61 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 ...
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1answer
22 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 $H^1(G)$ ...
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1answer
38 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 ...
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1answer
17 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) ...
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2answers
32 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\ ...
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0answers
18 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
14 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
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1answer
39 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 ...
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2answers
126 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 ...
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2answers
25 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 ...
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2answers
117 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
7 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 ...