# Tagged Questions

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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### Dilation and translation of the Dirac Delta distribution

Given $\delta(ax) = \frac{1}{|a|}\delta(x)$ and $\delta(ax-b) = \frac{1}{|a|}\delta(x-b/a)$ , it it true also that $\delta(a(x-b)) = \frac{1}{|a|}\delta(x-b)$ ?
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### The issue of treating an inverse Fourier transform in terms of a tempered distribution.

Consider the wave equation $$u_{tt}=\Delta{u} \quad u(x,0)=f(x) \quad u_t(x,0)=g(x) \tag{*}$$ A solution to this equation is given by $$u(.,t)=f*\partial_t\Phi_t+g*\Phi_t \tag{**}$$ where $\Phi_t$...
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### convolution -questions

I'm lost, can you help me please. How compute the product convolution between two distributions $T$ and $S$? (we suppose that $T * S$ exist)? How we compute the product convolution between an ...
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### Convolution computing

How we can compute the convolution product $$\Big(\sum_{n=0}^{+\infty} \delta_n^{(n)}\Big) \star \Big(\sum_{n=0}^{+\infty} \delta_n\Big)$$ where $\delta$ is Dirac distribution? Thank's for the help
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### Close fourier transforms implies close time domain functions?

What conditions do we need so that $A(f)\approx B(f) \Rightarrow \mathcal{F}^{-1}\left\{A\right\}(t)\approx\mathcal{F}^{-1}\left\{B\right\}(t)$ The Fourier transforms of my two things look alike. In ...
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### $f, \hat {f} \in L^{p} \cap L^{\infty} \implies f\in A(\mathbb R)$?

$1\leq p \leq \infty$. We put, $$X_{p}= \{f\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R) :\hat{f}\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R)\};$$ and we consider the algebra of Fourier ...
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### Strange Dirac delta distribution contradiction

Consider the following integrals in variables $x,y$ over the whole $\mathbb{R}$, where $a,b\in\mathbb{R}/0$ are constants: $$\int dx \int dy ~\delta(x-a)\delta(y-b\,x)=\int dy ~\delta(y-b\,a)=1$$ In ...
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### Expected distance within a distribution is smaller?

consider we have two general distributions $f_1$ and $f_2$, assume they have different support $S_1$ and $S_2$. Is the expected distance btween two points draw from the same distribution smaller than ...
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### How to prove a tempered distribution always has a order?

I just learn some distribution theory. In Friedlander's book, "Introduction to the theory of distributions" (page 97), he said: The dual of $\mathscr{S}(R^n)$ (Schwartz Space) consists of ...
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### Generalized functions and vector calculus theorems

To apply the divergence theorem, for example, there are conditions on your function. Your function must be a function in the ordinary sense in the first place. But in Electrodynamics, sometimes our ...
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I'd like to find: $$\lim_{\varepsilon\rightarrow 0}\frac{\varepsilon}{\varepsilon^2+x^2}\qquad \mbox{ in }\mathcal D'(\mathbb{R})$$ And I started with the definition: $$\left\langle \frac{\... 1answer 59 views ### the continuous functions with norm I'm having trouble trying to understand what does means the first expression in particular the last term in it should we add \|f\|_{\infty} \leq \infty or what i can't see what is his role (\|f\|... 2answers 146 views ### Uniqueness for tempered distributional Cauchy problems Question. Assume that U\in C^1(\,[0, \infty)\to \mathcal{S}'(\mathbb{R}^n)\,) is a solution to the following tempered distributional Cauchy problem$$\tag{CP}\begin{cases} \frac{ d U}{dt} = f \...
Given the following definition of the Dirac's Delta: $$\delta: \mathcal{D}(\mathbb{R}^n) \to \mathbb{R}: \varphi \mapsto \langle \delta,\varphi \rangle = \varphi(\mathbf{0})$$ where \$\mathcal{D}(\...