The integral of the convolution between functions f

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Discussion Overview

The discussion revolves around proving that the integral of the convolution between functions \( f \) and \( g \) equals the product of their integrals. It involves mathematical reasoning related to convolution and integration techniques.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant expresses difficulty in proving the statement about the integral of the convolution and seeks hints.
  • Another participant asks for clarification on what the original poster has tried and where they are encountering difficulties.
  • A third participant provides a mathematical expression involving Fubini's theorem, suggesting a step-by-step approach to the proof.
  • The original poster acknowledges the help received and reflects on a misunderstanding regarding variable translation in the context of integration over the entire real line.

Areas of Agreement / Disagreement

The discussion does not appear to reach a consensus, as participants are still exploring the proof and clarifying their understanding of the mathematical concepts involved.

Contextual Notes

There may be limitations related to assumptions about the functions \( f \) and \( g \), as well as the conditions under which the convolution and integrals are defined.

muzialis
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Hello there,

I am really struggling to prove that
"The integral of the convolution between functions f and gequals the product of their integrals", http://en.wikipedia.org/wiki/Convolution#Integration
Can anybody give me a hint?

Many thanks
 
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What did you try?? Where are you stuck?
 


∫{∫f(y)g(x-y)dy}dx = ∫f(y){∫g(y-x)dx}dy (Fubini)
= ∫f(y){∫g(u)du)}dy = ∫f(y)dy∫g(u)du
 


Thanks very muhc for your help.
I was following the line given by Mathman, but did not realize that the variable translation would not affcet the value of the integral as the integration domain is the whole real line, many thanks
 

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