- #1

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f✶g(x) = ∫

_{[0,x]}f(t)g(x-t)dt.

I would like to know if f✶g is a differentiable function of x.

If, for example, g(t) = 1 for t ≥ 0 then f✶g(x) = ∫

_{[0,x]}f(t)dt has a derivative equal to f(x). But what about in general?

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- Thread starter Zafa Pi
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- #1

- 631

- 132

f✶g(x) = ∫

I would like to know if f✶g is a differentiable function of x.

If, for example, g(t) = 1 for t ≥ 0 then f✶g(x) = ∫

- #2

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If either ##f## or ##g## is differentiable, then so is the convolution.

- #3

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Great, thanks. I now see that. But what happens if both f and g are nowhere differentiable?If either ##f## or ##g## is differentiable, then so is the convolution.

- #4

Samy_A

Science Advisor

Homework Helper

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Jarník, V. "Sur le produit de composition de deux fonctions continues." Studia Mathematica 12.1 (1951): 58-64

https://eudml.org/doc/216531

- #5

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Merci beaucoup. It seems that I am in good company since Mikusinski asked the question as well The question came to me as I was reading his "operational calculus". I worked on it for a day and gave up. Now I'll pour over the article. Thanks again.

Jarník, V. "Sur le produit de composition de deux fonctions continues." Studia Mathematica 12.1 (1951): 58-64

https://eudml.org/doc/216531

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