Odd/even functions and periodicity

[dimitri151]

You can prove if f(x) is an odd function and f(x+ t) is an even function then f(x) is periodic with period at most 4t. Are there other theorems like that?i know this is a somewhat open ended and general question, it's just i would like to squeeze some more results from this angle and can not.

 

[Simon Bridge]

i would like to squeeze some more results from this angle and can not.

... what "more results"? "more" suggests you have already got some results. What results have you got so far?

Consider the specific example where f(x)=sin(x).
The general approach would start with the definitions of both.

(modify slightly so that f(x-t) is even, makes it easier to write...)

if f(-x)=-f(x) and f(t-x)=f(x-t) then f(x)=f(x-nt): n in Z (?)

for f(x)=sin(x), t=pi/2, n=4.

 

[dimitri151]

The result so far is that if f(x) is odd and f(x+t) is even then f(n2t) =0 for all integer n, f(x) is periodic, and the minimum period is no greater than 4t.