The sum and multiplication of periodic functions

  • #1

Homework Statement



Hi, my question is whether the sum and multiplication of two periodic functions (with a common period) are periodic.
Our functions are R[itex]\rightarrow[/itex]R.

Homework Equations





The Attempt at a Solution


f(x)=f(x+T) g(x)=g(x+T) T is the period.
h(x)=f(x)+g(x)
h(x+T)=f(x+T)+g(x+T)=f(x)+g(x)=h(x)

Hence, h(x) is also periodic.
What I did is similar for multiplication. Is there any flaw in this? I searched a bit and found out this may not hold every time, but I guess that was about Fourier series, which I have no idea about.
Thanks for any hint :)

Homework Statement





Homework Equations





The Attempt at a Solution


Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
LCKurtz
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Your argument is fine. You only potentially run into difficulties when the periods aren't the same.
 
  • #3
Thanks, LCKurtz. Could you please explain what happens if the periods are not the same? Is it too complicated for a freshman in maths? :)
 
  • #4
LCKurtz
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Thanks, LCKurtz. Could you please explain what happens if the periods are not the same? Is it too complicated for a freshman in maths? :)
If you have periods like 2 and 3, then you need to use the least common multiple for the period of the sum. But if your two periods are 2 and [itex]\pi[/itex], there is no lcm and the sum isn't periodic at all. That happens when the ratio of the periods isn't a rational number.
 
  • #5
Thanks again :) Wow, I haven't thought of it before...
One last question, if you don't mind me :) What is an almost periodic function? It is a term I came across today, and would be grateful if you could explain this, too.
 
  • #6
LCKurtz
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Thanks again :) Wow, I haven't thought of it before...
One last question, if you don't mind me :) What is an almost periodic function? It is a term I came across today, and would be grateful if you could explain this, too.
You can read about that here:
http://planetmath.org/encyclopedia/AlmostPeriodicFunction.html [Broken]

and other links you can find with Google.
 
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