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Correct definition of convolution of distributions?

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Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Wikipedia states, that the definition of convolution of function 𝑓 𝑓 with a distribution 𝑇 𝑇 is ⟨𝑇∗𝑓,𝜑⟩=⟨𝑇, 𝑓 ̃  ∗𝜑⟩ ⟨ 𝑇 ∗ 𝑓 , 𝜑 ⟩ = ⟨ 𝑇 , 𝑓 ~ ∗ 𝜑 ⟩ where ⟨𝑇,𝑓⟩=𝑇(𝑓(𝑥)) ⟨ 𝑇 , 𝑓 ⟩ = 𝑇 ( 𝑓 ( 𝑥 ) ) and 𝑓 ̃  = d −1 𝑓(𝑥)=𝑓(−𝑥) 𝑓 ~ = d − 1 𝑓 ( 𝑥 ) = 𝑓 ( − 𝑥 ) dilation of 𝑓 𝑓 and this should hold ∀𝜑∈𝒮 ∀ 𝜑 ∈ 𝑆 . Then, convolution of distributions is defined by (𝑇∗𝑆)∗𝜑=𝑇∗(𝑆∗𝜑) ( 𝑇 ∗ 𝑆 ) ∗ 𝜑 = 𝑇 ∗ ( 𝑆 ∗ 𝜑 ) and 𝑇∗𝜑 𝑇 ∗ 𝜑 is supposed to be a function. My questi

Correct definition of convolution of distributions? - Mathematics Stack Exchange Stack Internal Knowledge at work Bring the best of human thought and AI automation together at your work. Explore Stack Internal Correct definition of convolution of distributions? Ask Question Asked 11 years, 5 months ago Modified 11 years, 5 months ago Viewed 11k times 10 $\begingroup$ Wikipedia states, that the definition of convolution of function $f$ with a distribution $T$ is $$\langle T\ast f,\varphi\rangle=\langle T,\tilde{f}\ast\varphi\rangle$$ where $\langle T,f\rangle=T(f(x))$ and $\tilde{f}=\mathrm{d}_

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