# Questions tagged [distribution-theory]

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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### How does derivative transfer to test function?

I set out to find a fundamental solution $E$ for the Poisson equation, i.e. a distribution $\mathscr D'(\Bbb R^d)$ such that $\Delta E = \delta$; I'm almost done. The only thing I have left to do is ...
1answer
89 views

### Show that $f(x) = e^x \cos(x)$ on $\mathbb{R}$ is a tempered distribution

As shown in the title. I know that the anti-derivative of $f(x) = e^x \cos(x)$ is $\frac{\sin(x)+\cos(x)}{2} e^x$, whose anti-derivative is $\frac{\sin(x)}{2}e^x$...but not sure how to prove it.
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1answer
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### find the limit in distributions(space of generalized functions)

How to find limit in $\mathcal{D}^{'}$ $$\exp(itx)(x+i0)^{-1}, t \rightarrow \infty$$ I try to use Sokhotsky's formula $(x+i0)^{-1} = -i\pi\delta(x) + \rho\frac{1}{x}$ , but did not come to a ...
1answer
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### Is it correct to set $\delta(t) f(t) = \delta(t) f(0)$ within a distribution?

I have some complicated probability distributions which come out as $$P(x,t) = \delta(t)G(x,t) + K(x,t),$$ where $G(x,t)$ and $K(x,t)$ are continuous in time. Is it permissible to simplify such ...
0answers
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### Poincaré Inequality on Gaussian Measures

So I have a working idea on Gaussian-Poincaré Inequality. Namely through the Ornstein-Ullenbeck Generator and Gaussian Integration by parts. Recently I have stumbled across Sobolev Spaces and have ...
0answers
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### Which distribution kernels represent continuous linear operators $\mathcal{D}(X)\to\mathcal{D}(Y)$?

If $X$ and $Y$ are smooth manifolds, which distribution kernels $K$ represent continuous linear operators $\mathcal{D}(X)\to\mathcal{D}(Y)$? According to Theorem VII in Schwartz's 1950 ICM paper, &...
1answer
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### Distributions dual to functions of polynomial growth.

The space of distributions on $\mathbb{R}^n$ is essentially found by requiring that it should be possible to apply the distribution to any bump function. Similarly, compactly supported distributions ...
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### $-\int_a^b F_{\mu}(x)\phi'(x)dx=\ldots=\int_a^b\phi(x)d\mu(x)$?

If I consider the interval $[a,b]$ and a positive measure $\mu$ on this interval, we can define a function $F_{\mu}(x)=\mu([a,x])$. I want to show that the distributional derivative of this function ...
0answers
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### Determine $p$ such that $u(x,y)=1-\max\{|x|,|y|\}\in W^{1,p}((-1,1)\times(-1,1))$

Let $\Omega$ be the open square $(-1,1)^2\subset\mathbb{R}^2$, $u\in L^1(\Omega)$ such that for each $(x,y)\in\Omega$, $u(x,y)=1-\max\{|x|,|y|\}$. Determine the weak gradient of $u$ and find $p$ such ...
1answer
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### Determine $a$ and $p$ such that $u(x)=|x|^{-a}\in W^{1,p}(B_1(0))$

Let $\Omega=B_1(0)=\{x\in\mathbb{R}^N:|x|<1\}$ and let $u\in L^1(\Omega)$ such that $u(x)=|x|^{-a}$, with $0<a<N$. Determine $\nabla u$ as distributional derivative. Then determine $a$ and $p$...
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### Zero distributions

A distribution, by the definition given in Shearer and Levy, is a function $f \in D'(\mathbb{R})$ with $f: C^{\infty}_c(\mathbb{R})=D(\mathbb{R}) \rightarrow \mathbb{R}$ s.t. $f$ is linear $f$ is ...
2answers
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### Product rule of distribution by multiplicating a $C^\infty$ function

I am reading S. Kesavan's book "Topics In Functional Analysis and Application" and trying to prove the product rule, can you check where I am going wrong? Let $\psi \in C^\infty (\mathbb{R})$...
1answer
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### Function which integrates to $0$ against test functions with mean $0$ is constant almost everywhere.

Suppose $U$ is a bounded domain in $\mathbb{R}^n$ and $u \in L^1(U)$ has the property that $$\int_{U}u\phi=0$$ for all $\phi \in C_{c}^{\infty}(U)$ which satisfy $\int \phi = 0$. I'd like to show ...
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### Prove $u, \Delta u \in L^2(\mathbb{R}^n)$ then $u \in H^2(\mathbb{R}^n)$

Let $k=2$ throughout. Let $u, \Delta u \in L^2(\mathbb{R}^n)$ then $u \in H^k(\mathbb{R}^n)$ where $H^k= \{u \in S'(\mathbb{R}^n): (1+|\xi|^2)^{\frac{k}{2}} \widehat u \in L^2\}$. I am a bit unsure ...
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### Delta function singularity

I am reading the Wiki article about singularities and I was wondering what kind of singularity is the Dirac delta function not defined as a distribution but as this way: \begin{equation} \delta^{\...
0answers
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### $-\Delta u+\epsilon \sin u=f$ has a unique solution in the sense of distribution

It's the last question in my exam. Prove that there is a neighborhood of $0$, denoted by $[-\delta,\delta]$, $\forall \epsilon \in [-\delta,\delta]$ the following equation has a unique solution in ...
1answer
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### Show that if $g \in \mathcal{D'}(\mathbb{R})$ satisfies $\frac{dg}{dx}=0$ then g is constant.

I'm having trouble figuring this out and its been bothering me for a few weeks now. The solution constructs a test function in a particular way to prove the result, but I have trouble understanding ...
1answer
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### Let $(a,b)$ and $f\in L_{\text{loc}}^1(a,b)$. For $x_0\in (a,b)$, $F(t)=\int_{x_0}^{t}f(s)ds$. Prove that, $DF=f$ (towards theoretical distribution).

Let $(a,b)$ and $f\in L_{\text{loc}}^1(a,b)$. For $x_0\in (a,b)$, consider $$F(t)=\int_{x_0}^{t}f(s)ds.$$ Prove that, $DF=f$ (towards theoretical distribution). I thought of the following: Let \$\...
0answers
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### How to define the grad, div and the curl of a distribution?

I have learned how to define the derivative or partial derivative of a distribution, but I still can't find a clear definition of the grad, div and curl of a distribution, I would appreciate it if you ...