# Tagged Questions

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### Absolute continuity and convolution

Suppose that $\mu$ is a finite Borel measure on the real line, $f, g\in L^1(\mu)$. Define $\nu=\mu\ast\mu$. Do I understand correctly that the convolution $f\mu\ast g\mu$ is absolutely continuous wrt ...
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### Proof commutativity of (differential) convolution operater

I tried to proof a claim and I'm not sure if I did it right. It would be great if someone could have a look at it! First I give a definiton: Let $h : [0, \infty ) \rightarrow \mathbb{R}$. We define ...
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### Proof of a corollary about associativity of (differential) convolution operater

I m working on the proof the following corollary for ages... I would appreciate any help!! Cor: Let $h : [0, \infty ) \rightarrow \mathbb{R}$. We define the convolution operator $*$ for the ...
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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 of a probability measure with a smooth function

If $f\in L^1(\mathbb{R}^n)$ and $g\in L^p(\mathbb{R}^n)$ then by Young's convolution inequality we have the estimate: $$\|f*g\|_{L^p}\leq \|f\|_{L^1}\|g\|_{L^p}.$$ Question: Let $\mu$ be a ...
### $f$ is bounded and continious $\Rightarrow$ the convolution integral $\int f(\tau)g(x-\tau)\text{ d}\tau$ is bounded and continuous
Let $g\in L^1(\mathbb{R}^n)$ and $f:\mathbb{R}^n\to\mathbb{R}$ be bounded and continuous. Why is the convolution integral $$f*g:\mathbb{R}^n\to\mathbb{R}\;,\;\;\;\int f(\tau)g(x-\tau)\text{ d}\tau$$ ...