# Tagged Questions

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### What is $D\delta$ if $D$ is ordinary differential operator and $\delta$ is the Dirac distribution?

I'm reading some material about single-variable distribution theory. More specifically, I was checking some theorems of the convolution algebra $\mathcal{D}_+$, where $\mathcal{D}_+$ is the space of ...
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### Prove or disprove: $e^{-nG(x)}$, normalized, is an approximation to the identity for $G(x)$ strictly convex

We are given the sequence of functions $$\phi_{n} = \frac{e^{-nG(x)}}{\int_{\mathbb{R}}e^{-nG(x)}dx}$$ for a nonnegative, strictly convex function $G$ (that is, $G'' \geq c$ for some $c>0$) that ...
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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 ...
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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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### Convolution of functions and measures

I need some help with this exercise. I'm not sure how to deal with it: Let $f(x)=e^{-x^2}$, $\mu$ the Lebesgue measure in $[0,1]$ and $\nu$ the Lebesgue measure in $[2,\infty)$. I have to find the ...
I need some help with this exercise: It proposes to show that convolution of distributions is not associative: If $T=T_1$ (distribution given by f=1), $S=\delta'$, and $R=T_H$ (we denote as $H$ the ...
I am trying to understand the following paper. In page 1191, in the beggining of the proof of Theorem 2.9. the authors consider the convolution $$v=\Gamma\star f$$ They claim that $v\in ... 1answer 320 views ### Convolution of distributions. We are given with distributions$f,g \in D'(\Bbb R)$. If$suppf\subset (-\infty,a)$and$supp(g)\subset(b,\infty)$then prove that$f*g$is well defined distribution. where$a$and$b$are real ... 1answer 119 views ### The differentiability of convolutions Yes, again, this type of question. Similar ones this and this. I come with another variant. Let$f\in\mathcal{S}$, i.e. Schwartz function, and$g\in L^{p}(\mathbb{R}^d),p\in[1,\infty]$. The following ... 2answers 103 views ### Problem of convolution. If we are given with a polynomial$\mathcal P$and a compactly supported distribution$g$. Can we prove that their convolution will be a polynomial again? 1answer 101 views ### convolution-distributions We denote by$E'(\mathbb{R})$the set of distribution with compact support , and$\mathcal{D}(\mathbb{R})$is the set of function$\mathcal{C}^{\infty}$with a compact support. 1) I want to compute ... 1answer 135 views ### Convergence of convolution of$L^p$function with a sequence of distributions let$h_n\in C_c^\infty (\mathbb{R}^d)$s.t.$\int h_n dm = 1$and$\operatorname{supp}(h_n)\to {0}$. I've proven that$h_n\to\delta_0$in$\mathcal{D}'(\mathbb{R}^d)$, now I'm trying to show that for ... 1answer 231 views ### Convolution between two distributions I want to define the convolution$*$between two distributions$S$and$T$. For a test function$\varphi$, can I say: $$\langle S * T, \varphi \rangle \doteqdot \langle S, T*\varphi \rangle$$ where ... 1answer 147 views ### how to compute the convolution of two measures explicitly Here is my example:u and v are the surface measures on the spheres {${x;|x|=a}$} and {${x;|x|=b}$} in$\mathbb{R}^{3}$.Then what's$u\ast v$? And what if in$\mathbb{R}^{n}$? 1answer 641 views ### Fourier transform of convolution of sinusoidal signals, or product of distributions (generalized functions) I will unashamedly say that this was at least spurred by homework. However I have gone far beyond the syllabus of the course and still can't find an authoritative answer. And it seems an interesting ... 1answer 154 views ### Verify this distribution convolution:$E(t,x)\ast (g(x)\delta(t)) = t\int_{\omega\in S^2}{\frac{g(x-t\omega)}{4\pi}dS(\omega)}\$
In our class notes we are asked to verify the following equality: $$E(t,x)\ast (g(x)\delta(t)) = t\int_{\omega\in S^2}{\frac{g(x-t\omega)}{4\pi}dS(\omega)}$$ where ...