The sum and multiplication of periodic functions

[life is maths]

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\rightarrowR.

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 :)

 

[LCKurtz]

Your argument is fine. You only potentially run into difficulties when the periods aren't the same.

 

[life is maths]

Thanks, LCKurtz. Could you please explain what happens if the periods are not the same? Is it too complicated for a freshman in maths? :)

 

[LCKurtz]

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 \pi, 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.

 

[life is maths]

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.

 

[LCKurtz]

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

and other links you can find with Google.