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The sum and multiplication of periodic functions

  1. Nov 14, 2011 #1
    1. The problem statement, all variables and given/known data

    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.

    2. Relevant equations



    3. 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 :)
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 14, 2011 #2

    LCKurtz

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    Your argument is fine. You only potentially run into difficulties when the periods aren't the same.
     
  4. Nov 14, 2011 #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? :)
     
  5. Nov 14, 2011 #4

    LCKurtz

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    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.
     
  6. Nov 14, 2011 #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.
     
  7. Nov 14, 2011 #6

    LCKurtz

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    You can read about that here:
    http://planetmath.org/encyclopedia/AlmostPeriodicFunction.html [Broken]

    and other links you can find with Google.
     
    Last edited by a moderator: May 5, 2017
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