Suppose that f is -periodic and let a be a fixed real number. Define
Show that F is 2\pi-period iff
My proof so far:
Assume that F is 2\pi-periodic, then F(x) = F(x+2pi) = ...
By a theorem, we know that
So
(We also need the condition that is integrable. Can you think of an example when this fails?)
I start you off.
1)Assume is integrable and -periodic.
2)We will show the forward direction, i.e. is -periodic implies .
3)Say that is a -periodic function.
4)By definition it means .
5)Now #4 is true for all , pick .
6)Thus,
7)By definition of we see that .
8)Thus, .
9)Since is -periodic we use the theorem that .
10)By #9 we see that .
Q.E.D.