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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1answer
36 views

Derivative as distribution

We consider the Heaviside function $H(x)$. $H'(0)$ doesn't exist. The derivative exists if we define $H$ as a distribution $$H: \phi \to \int_{-\infty}^{+\infty} H(x) \phi(x) dx= \int_0^{+\infty} \...
8
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2answers
99 views

Can a “continuous” convex combination not be element of the convex hull?

Short version of question: can a "continuous" convex combination not be element of the convex hull? I am not a mathematician, so please excuse me if I am not precise. I consider first, e.g., 4 ...
1
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2answers
67 views

Is the function $|x|$ in $W^{1,p}$?

I have the following question: We consider in the segment $I=]-1,1[$, the function $f(x)=|x|.$ The question is: For each value $p \in [1,+\infty[$ do we have $f \in W^{1,p}(I)$? My purpose is: We ...
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0answers
15 views

Understanding an example of Renardy's book (example 5.48)

In the example $5.48$, authors say that: To prove that a sequence of integrable functions $f_n: \mathbb{R} \to \mathbb{R}$ converges to the delta function, it suffices to show that the primitives ...
0
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1answer
26 views

Show that $u_{tt}=u_{xx}$ in the sense of Distributions

Let $u(x,t)=f(x+t)$ , where $f$ is any locally integrable function on $\mathbb{R}$. Show that $u_{tt}=u_{xx}$ in the sense of Distributions My try: For $\phi \in D(\mathbb{R})$, $$(u_{tt},\phi)=-(...
0
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1answer
25 views

Describe the distributional derivative of $f$

Let $f$ be a piece wise defined function with piece wise continuous derivative. Describe the distributional derivative of $f$. My try: If I suppose that the jump discontinuities are at the points $-...
1
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0answers
18 views

Distribution functions: differentials in the numerator or denominator

One paper I'm looking at says, $n(M, z) \, dM \, dz$, the number of sources with mass $M$ at a redshift $z$, in the mass interval $dM$ occurring in the redshift interval $dz$. While another ...
1
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1answer
45 views

Example of a linear functional, but not a distribution

I'm looking for an example of a linear functional $u: C_c^\infty(\Omega) \to \mathbb C$ ($\Omega \subset \mathbb R^n$ open), which is not a distribution. I could not find anything... I thought of ...
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0answers
20 views

Distributions on compact and semi-open intervals

In the theory of distributions (aka generalized functions), one considers mostly distributions $T \in \mathcal{D}(\Omega)$ on an open subset $\Omega \subseteq \mathbb{R}^n$. Hereby, the space $\...
2
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1answer
81 views

Derivative of Dirac delta function as a measure

Dirac delta function can be defined in several ways. I know two definitions. One is as a distribution and the other is as a measure. I found many materials on the derivatives of delta function as a ...
2
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0answers
39 views

Uniform convergence of compactly supported function

If $f\in C_c^\infty$, i.e. Compactly supported, then for $g_k(x)=f(kx)$, would $g_k\to 0$ uniformly as $k\to \infty$ since g vanishes outside a given compact set which won't be touched for $k$ ...
0
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1answer
45 views

existence of smooth functions with support $0$

In my real analysis homework, theres a question which begins with "let $F$ be a distribution on $\mathbb{R}^n$ such that $\operatorname{supp}(F)=\{0\}$" This means that the only $\phi \in C_c^\...
0
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0answers
141 views

Folland real analysis 9.11

This comes from question 9.11 of Folland's Real analysis textbook. Unfortunately, I have no idea to how to start with this question. So can some one help me with part $a$? For part $a$, I can not ...
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0answers
21 views

Behaviour of functions in weighted sobolev spaces

If $f$ and $Df$ are in $L^2(\mathbb{R}, e^{u^2} dx)$, can we say $f(u)e^{\frac{u^2}{2}}$ is bounded. Here $Df$ distributional derivatie of $f$. That is, If $\int_{\mathbb{R}} \lvert f(u) \rvert^2 e^{...
1
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1answer
97 views

Proof of regular version of the Urysohn lemma

I know it's a well-known result, but I have not found any clear formalization, and I need a clear formalization. So I want to know if you agree with this formalization, and this proof. Thank you for ...
1
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0answers
54 views

topology of $C^\infty _K(\Omega)$

Let be $\Omega \subset \mathbb R^n$ and open set. The space of smooth function $C^\infty (\Omega)$ is endowed with the topology generated by the seminorms $\{p_{K,m}: \;K \subset \Omega \;\text{...
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0answers
20 views

Is this distribution regular?

I have to check whether the following distribution is regular: a) $$ [|3x-1||_{[0, \infty]}]'' $$ b) $$ [|3x-1||_{[-\infty,0]}]'' $$ I've got that: a) $$[|3x-1||_{[0, \infty]}]''(\Phi) = -3\Phi(0)...
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0answers
34 views

Prove that Heaviside's function H(x) is a solution for equation

Prove that Heaviside's function H(x) is a solution for equation $$ \frac{\partial^2 u}{\partial x_1\partial x_2}=0 $$ $$$$ $$$$ For example, to solve that function $$ F(x)=e^{2x}H(x)$$ is solution ...
4
votes
1answer
105 views

A commutation between curl and integral

I have been struggling to understand the only derivation of Ampère's law from the Biot-Savart law for a tridimensional distribution of current (which, needless to say, is not the case of a linear ...
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0answers
40 views

Topology for distributions on a compact space

I'm having trouble in distribution theory, though not in the usual setting. The context is in theoretical physics, trying to solve BF theory. My goal is to solve the following equation: $$\forall i \...
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1answer
23 views

Show that $\phi_{y,\epsilon} \in D(\mathbb{R^m})$

For a fixed $y$ $$\phi_{y,\epsilon}(x)= \left\{ \begin{array}{ll} \exp(-\frac{\epsilon^2}{\epsilon^2-|x-y|^2}) & \mbox{if $|x-y| \lt \epsilon$};\\ 0 & \mbox{otherwise}.\end{...
0
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1answer
43 views

Homogeneous distribution

In Wikipedia, it says The Dirac delta function is homogeneous of degree −1, with the following formula: However, I can not understand why the last equality is true. Can someone show me the detailed ...
0
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1answer
30 views

Is this operator a distribution?

Is this operator: $$T: \mathcal{C}^{\infty}_0 \ni \varphi \to \lim_{x \to \infty} x^2 e^{-x} \varphi'(x) \in \mathbb{R}$$ a distribution (generalized function)? I need to check two things: whether ...
1
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0answers
27 views

To what Sobolev space does this function belong to?

I am given this function: $$f(x) = e^{- \sqrt{|x|}}$$ and I want to find $k\in \mathbb{N}, \ p \ge 1$ such that $f \in W^{kp} (\mathbb{R})= \{ f \in L^p (\mathbb{R}) \ | \ \forall \alpha \le k: \ D^{\...
0
votes
1answer
104 views

Solution of this definite integral?

I want to find the expression for the following integral $$\int_0^\infty\text{d}x\frac{e^{i k x}}{x}$$ I have tried deriving with respect to $k$, transforming into an integral over the whole real ...
0
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0answers
40 views

show that $f_{\epsilon} \in D(\Omega)$; moreover, $f_{\epsilon} \to f$ uniformly as $\epsilon \to 0$.

Let $K$ be a compact subset of $\Omega \subset \mathbb{R^m}$, $\Omega$ is open and nonempty and let $f \in C(\Omega)$ have support contained in $K$. For $\epsilon \gt 0$, let $$f_{\epsilon}(x)=\frac{...
3
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1answer
28 views

Show that $f_{n}^2(x)$ does not converge in $D^1({\Omega})$

Let $$ f_n(x) = \left\{ \begin{array}{ll} n & \mbox{if $0 \lt x \lt \frac{1}{n}$};\\ 0 & \mbox{otherwise}.\end{array} \right. \ $$ I have to show that $\lim_{n \to \...
0
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0answers
24 views

Show that any two different functions in $C(\Omega)$ differ as generalized functions as well.

Show that any two different functions in $C(\Omega)$ differ as generalized functions as well. Let $f,g \in C(\Omega)$. Since $f\ne g$, there is a $x_0 \in \Omega$ such that $f(x_0)\ne g(x_0)$. Hence ...
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0answers
27 views

Assume that every point in $G$ has a neighborhood on which $f$ vanishes. Prove that $f$ vanishes on $G$

Let $f$ be a generalized function on $\Omega$. Let $G$ be an open subset of $\Omega$. Assume that every point in $G$ has a neighborhood on which $f$ vanishes. Prove that $f$ vanishes on $G$ Let $\phi ...
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0answers
20 views

What it means for a generalized function to be periodic or radially symmetric??

Let $T$ be a generalized function. I need to provide definitions for $T$ to be periodic and radially symmetric. A function (on $\mathbb{R})$is said to be periodic if there exists a $p \in \mathbb{R}$...
1
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2answers
92 views

What is the definition of the order of a distribution?

A linear functional $T$ on $\mathcal{D}(\Omega)$ is a distribution if $\phi_n \to 0$ in $\mathcal{D}(\Omega)$ $\Rightarrow$ $T(\phi_n) \to 0$ in $\mathbb{R}$. But I cannot find what the order of a ...
1
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1answer
21 views

Tempered representatives of a special class of distributions

Suppose that a distribution $R\in D'(\Bbb R)$ satisfies the following estimation for an independent constant $c$: $$\forall \phi\in D(\Bbb R)\quad |\langle R,\phi\rangle|\le c\|\phi, \,L^1(\Bbb R)\|....
1
vote
1answer
47 views

Bump functions converging to an indicator

Suppose $K\subset\mathbb{R}^n$ has a smooth boundary, and let $\phi_s(x)$ be bump functions converging pointwise to the indicator of $K$, i.e. $$\underset{s\rightarrow\infty}{\lim}\phi_{s}(x)=\mathbf{...
1
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1answer
58 views

limit of a distribution

Show that $$\lim_{\epsilon \to 0^{+}} \left\langle \frac{\epsilon}{x^{2}+ \epsilon} ,\phi \right\rangle =\langle \delta,\phi\rangle $$ where $\phi\in D(\mathbb{R}) $ and $ \frac{\epsilon}{x^{2}+ \...
0
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0answers
26 views

convolution properties of distributions

Let $f,g,h \in D'(R^n)$. How we define the convolution of these functions? I'm trying to show some properties of convolutions such as $\delta\ast f=f$ $(f\ast g)' = f'\ast g=f\ast g'$ $(f\ast g) \...
3
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0answers
63 views

Fourier distribution $\frac{e^{i|x|}}{|x|}$

I need help to calculate Fourier transform in distribution sense of $\frac{e^{i|x|}}{|x|}$ in $D'(\mathbb{R}^3)$ we have $ \frac{e^{i|x|}}{|x|} \in L^1_{loc}(\mathbb{R}^3)$ edit, Let $E(x)=\frac{e^{i|...
0
votes
1answer
46 views

Borel lemma : wikipedia proof

In the proof of Borel's lemma, I don't understand why we use $\psi\left(\frac{t}{\epsilon_m}\right)$ for a sufficient small $\epsilon_m$ and not $\psi(t\cdot \epsilon_m)$, as you need to keep ...
1
vote
1answer
27 views

Prove order of distribution $\Lambda_{1/x}$ is 1

The distribution is defined as: $$\Lambda_{1/x}(\varphi)=\lim_{\varepsilon\rightarrow0+}\int_{\mathbb{R}\backslash(-\varepsilon,\varepsilon)}\frac{\varphi(x)}{x}\ \mathrm{d}x$$ I tried integrating ...
3
votes
1answer
54 views

Computing the limit of an integral

Consider the following integral $$ \int_{-\infty}^{\infty}f(t) K(\frac{a-t}{h})dt $$ where (1) $h>0$, $a \in \mathbb{R}$ (2) $f:\mathbb{R}\rightarrow[0,\infty)$ is such that $\int_{-\infty}^{\...
2
votes
1answer
39 views

Given a $C_c^∞(G)$-valued random variable, is $C_c^∞(G)∋φ↦\text E[\langle\xi,φ\rangle]$ an element of the dual space of $C_c^∞(G)$?

Let $G\subseteq\mathbb R^d$ and $$\mathcal D:=C_c^\infty(G)$$ be equipped with some topology $\tau$ $\mathcal D'$ be the dual space of $\mathcal D$ and $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the ...
0
votes
1answer
29 views

What is a generalized stochastic process? I've found two different definitions. Are they equivalent?

Let $\mathcal D:=C_c^\infty(\mathbb R^d)$ and $\mathcal D'$ be the dual space of $\mathcal D$. What is a generalized stochastic process? I've found two different definitions in some textbooks: ...
1
vote
1answer
29 views

How to show that $\delta^3(\vec{x}-\vec{a})=\lim_{\alpha\to0} \frac{\alpha/\pi^2}{(\alpha^2+|\vec{x}-\vec{a}|^2)^2}$

I hope to show that: $$\delta^3(\vec{x}-\vec{a})=\lim_{\alpha\to0} \frac{\alpha/\pi^2}{(\alpha^2+|\vec{x}-\vec{a}|^2)^2}$$ I want to show by: $$\int^{+\infty}_{-\infty} f(\vec{x}) \lim_{\alpha\to0} \...
1
vote
2answers
29 views

functions acting as linear functionals on their dual space

Supposing $f\in L^p$, where p and q are conjugate exponents, what does it mean that "f is completely determined by its action as a linear functional on $L^q$"? (Quoting Folland's Real Analysis here). ...
0
votes
0answers
14 views

Why can “sufficiently smooth” distributions of $C_c^\infty([0,\infty)\times G)$ be represented as functions of $t\in [0,\infty)$ and $x\in G$?

Let $G\subseteq\mathbb R^d$ be a bounded domain and $$\mathcal D:=C_c^\infty([0,\infty)\times G)\;.$$ Assuming that $\mathcal D$ is equipped with the usual locally convex topology, the space of ...
3
votes
1answer
135 views

$\nabla^2(\|\boldsymbol{x}-\boldsymbol{x}_0\|^{-1})=-4\pi\delta(\boldsymbol{x}-\boldsymbol{x}_0)$ with distributions defined on Schwartz space

I know, from a recent enlightening answers received here, that, if we define the distribution represented by Dirac's $\delta$ on the space $K$ of test functions of class $C^\infty$ whose support is ...
7
votes
2answers
182 views

Dirac's delta in 3 dimensions: proof of $\nabla^2(\|\boldsymbol{x}-\boldsymbol{x}_0\|^{-1})=-4\pi\delta(\boldsymbol{x}-\boldsymbol{x}_0)$

If $T_f$ is a distribution, i.e. a linear functional, continuous according to the convergence defined here, defined on the space $K$ of the functions of class $C^\infty$ that are null outside a ...
2
votes
1answer
45 views

What's the distributional derivative of a Banach space valued almost surely continuous stochastic process?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $\lambda$ be the Lebesgue measure on $[0,\infty)$ $(H,\left\|\;\cdot\;\right\|)$ be a Banach space over the field $\mathbb F\in\...
0
votes
1answer
72 views

Does the weak divergence exist for each $\mathcal L^2(\Omega;\mathbb R^d)$-function?

Let $\Omega\subseteq\mathbb R^d$ be open. $v:\Omega\to\mathbb R$ is called weak divergence of $u:\Omega\to\mathbb R^d$ $:\Leftrightarrow$ $$\int_\Omega v\varphi\;{\rm d}\lambda=-\int_\Omega\langle ...
2
votes
1answer
58 views

Prove that $\frac{t}{t^2-1}$ is a tempered distribution

I want to compute the Fourier transform of $\frac{t}{t^2-1}$, and in order to do so I need to prove in which space is the function. Clearly the function is not $L^1(\mathbb{R})$ neither $L^2(\mathbb{R}...
3
votes
2answers
52 views

Why is $\frac{d^2}{dx^2} e^{-|x|} = e^{-|x|} - 2 \delta(x)$

How can you show that $$\frac{d^2}{dx^2} e^{-|x|} = e^{-|x|} - 2 \delta(x) ? $$ I found this result using Wolfram Alpha and it seems strage to me, how the delta function appears here ...