# 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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### Why are the following limits used?

exercise in general says Find k if the joint probability density of X, Y, and Z is given by $$f(x,y,z)=kxy(1−z) \text{ for } 0<x<1, 0<y<1, x+y+z<1$$ then the answers define the ...
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### A problem with Lebesgue integral in functional analysis

Let $F \in D '$ have a compact support, $F(\varphi)\geq0$ for any $\varphi \geq0$. How can i prove that $F(\varphi) = \int \! \varphi \, \mathrm{d}\mu$ for some non-negative measure $\mu$?
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### Does the extension of an element of $W^k$ by $0$ still lie in $W^k$?

Let $U\subset\mathbb{R}^n$ be an open set and $Z$ a closed subset of $U$ . We denote by $W^{k}(U)$ the Sobolev space of functions whose derivatives (in the sense of distribution theory) up to order $k$...
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### Limit in distributions of $\frac{\sin(tx)}{x}$

How can I find the limit of $\frac{\sin(tx)}{x}$ as $t \to \infty$ in $D'$ ? I understand that i need to see the $\lim_{t \to \infty}{\int_{\infty}^{\infty}{\frac{\sin(tx)\phi(x)}{x}dx}}$ for every ...
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### Distributional equation $p(x) T = \varphi(x)$, $p$ polynomial, $\varphi$ smooth with rapid decrease

I am posing a variant of this question. Data: a polynomial $p(x)$, $x\in\mathbb{R}$, with complex coefficients, and function $\varphi:\mathbb{R}\to\mathbb{C}$, smooth, with rapid decrease (say, ...
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### Proof that a zero-variance Gaussian function becomes a Delta distribution

Consider the Gaussian function: $$f_{N(\mu, \sigma^2)}(t) = \frac{1}{\sigma \sqrt{\pi}} \exp \left[ {-\frac{1}{2}\left( \frac{t - \mu}{\sigma} \right)^2} \right]$$ I have seen in some texts ...
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### Solving the “Transport” PDE in the sense of distributions with Dirac Delta Source

Let $\delta_0$ be the standard Dirac Delta distribution. I wish to solve the PDE $$u_t+cu_x=\delta_0$$ in the sense of distributions with initial condition $u(x,0)=g(x)$ for some continuous $g$. That ...
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### Making sense of covariance of space-time white noise as a product of delta distributions

The covariance of space-time white noise $\dot{W}(x,t)$ is given by $\mathbb{E}\dot{W}(x,t)\dot{W}(y,s) = \delta(t-s)\delta(x-y)$, where the $\delta$-distribution satisfies $\delta(x) = 0$ if $x\neq 0$...
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### What is the formal definition of the singular support of a distribution?

The definition I have is that：For a distribution $u \in \mathcal{D}'(U)$ where $U$ is an open subset of $\mathbb R^n$, a point $x$ is in the singular support of $u$ if $u$ is not smooth on an open set ...
### Show that $\langle T, \varphi \rangle$ is a tempered distribution
Show that $\langle T, \varphi \rangle = \sum_{n=1}^{\infty} \varphi(n)$ is a tempered distribution on $\mathbf{R}^1$. My question is from Strichartz book, A guide to Distribution theory and Fourier ...