Is Convolution and Integration: A True Equality in Mathematics?

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Jhenrique
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I was verifying that:

Equation: integral over R^n of (f * g)(x) dx equals the product of the integral over R^n of f(x) dx and the integral over R^n of g(x) dx.


http://en.wikipedia.org/wiki/Convolution#Integration


But, I'd like to know if this equality is true:
[tex]\int f(x)g(x)dx \overset{?}{=} \int f(x)dx \ast \int g(x)dx[/tex]

Subquestion: this theorem above can be applied in summation too?
[tex]\sum_{x}\left (f(x)\ast f(x) \right )\overset{?}{=}\left(\sum_{x} f(x)\right)\left (\sum_{x} g(x) \right )[/tex]
[tex]\sum_{x}f(x)g(x) \overset{?}{=} \sum_{x}f(x) \ast \sum_{x}g(x)[/tex]
Thanks!
 

Discussion

Jhenrique said:
But, I'd like to know if this equality is true:
[tex]\int f(x)g(x)dx \overset{?}{=} \int f(x)dx \ast \int g(x)dx[/tex]

The easy way to check is differentiate both sides. What do you know about the derivative of the convolution.
 
Jhenrique said:
But, I'd like to know if this equality is true:
[tex]\int f(x)g(x)dx \overset{?}{=} \int f(x)dx \ast \int g(x)dx[/tex]

But ##\int f(x) dx## and ##\int g(x)dx## are just real numbers, no? What is the convolution of two numbers?
 
pwsnafu said:
The easy way to check is differentiate both sides. What do you know about the derivative of the convolution.

That:
Equation: d/dx (f * g) = df/dx * g = f * dg/dx, where * denotes convolution, showing the derivative of a convolution of f and g.
 
R136a1 said:
But ##\int f(x) dx## and ##\int g(x)dx## are just real numbers, no? What is the convolution of two numbers?

No! The result is a function!

pwsnafu said:
The easy way to check is differentiate both sides. What do you know about the derivative of the convolution.

Results: f(x) g(x) = f(x) * ∫ g(x) dx

But, my ask remains! My as ask now is f(x) g(x) is equal to f(x) * ∫ g(x) dx ?
 
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