Differentiability of convolution

In summary, the convolution of two continuous functions on the right half-line, [0,∞], is defined by f✶g(x) = ∫[0,x] f(t)g(x-t)dt. If either ##f## or ##g## is differentiable, then so is the convolution. However, there exists a continuous function x for which the convolution x*x is only differentiable in 0, as shown in the 1951 paper by Jarník. This question was also posed by Mikusinski in his "operational calculus".
  • #1
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If f and g are continuous functions on the right half-line, [0,∞], then f✶g, the convolution of f and g, is defined by
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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  • #2
If either ##f## or ##g## is differentiable, then so is the convolution.
 
  • #3
micromass said:
If either ##f## or ##g## is differentiable, then so is the convolution.
Great, thanks. I now see that. But what happens if both f and g are nowhere differentiable?
 
  • #4
The following paper from 1951 constructs a continuous function x, for which the convolution x*x is only differentiable in 0.

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

https://eudml.org/doc/216531
 
  • #5
Samy_A said:
The following paper from 1951 constructs a continuous function x, for which the convolution x*x is only differentiable in 0.

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

https://eudml.org/doc/216531
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.
 

1. What is the definition of differentiability of convolution?

The differentiability of convolution refers to the ability to take the derivative of a convolution function. It is a measure of how smoothly the output of a convolution will change when the input is changed. A convolution is said to be differentiable if its derivative exists at every point in its domain.

2. Why is differentiability of convolution important?

Differentiability of convolution is important because it allows us to analyze the behavior of convolution functions and to solve complex problems involving them. It also allows us to use techniques such as gradient descent to optimize the parameters of a convolutional neural network.

3. How is differentiability of convolution related to the concept of continuity?

Continuity and differentiability are closely related concepts in calculus. A function is said to be continuous if it has no abrupt changes or breaks, and it is differentiable if it has a well-defined derivative at every point. Therefore, a function that is differentiable at a point is also continuous at that point.

4. Can a convolution be differentiable at some points but not others?

Yes, it is possible for a convolution to be differentiable at some points but not others. This is because the differentiability of a convolution depends on the smoothness of the function being convolved and the properties of the convolution kernel. In some cases, the convolution may have discontinuities or sharp changes that make it non-differentiable at certain points.

5. How can we determine if a convolution is differentiable?

To determine if a convolution is differentiable, we can use the rules of differentiation to find the derivative of the convolution function. If the derivative exists at every point, then the convolution is considered differentiable. Additionally, we can also use the concept of continuity to check if the convolution is differentiable at a specific point.

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