Register to reply 
Function with two periods 
Share this thread: 
#1
Jan2207, 12:58 PM

Emeritus
Sci Advisor
PF Gold
P: 4,500

Suppose a function f has two periods a, b with a =/= n*b or vice versa. Then is f necessarily a constant? What if a is not a rational multiple of b? Answers involving whether f is continuous or not are appreciated. Basically, here's now I started:
WLOG, suppose b > a. Then ba < b, and either a < ba or a > ba. Either way, we've found a third period which does not satisfy the integral (or rational) multiplicity requirement. Since one of these periods is smaller than the other, we can subtract the smaller from the larger to get another smaller period. This process should be repeatable infinite times to get infinite discrete periods which tend to zero. But what is this enough for? EX: Suppose p is Pi and e is, well, e. Then pe is a period, and e(pe) = 2ep is a period. 2ep  (pe) = 3e2p is a period, and so is 3e 2p + (2ep) 5e3p. So you can make a whole bunch of periods.... I don't know which of the periods I have listed here is the smallest (I'm too lazy to get a calculator out and do this for more than a couple runs), but I think you can see the process of shrinking the period. So you'll get periods of the form k_{i}*a  m_{i}*b, where, I believe, k_{i}/m_{i} approaches b/a. Thoughts? 


#2
Jan2207, 01:54 PM

Sci Advisor
HW Helper
P: 2,020



#3
Jan2207, 02:52 PM

HW Helper
P: 3,348

Ahh Im not too sure about anything, but i was wondering how a function could have to periods, and then I realised the most obvious answer of extending in more diemensions :)



#4
Jan2207, 08:49 PM

Sci Advisor
P: 2,751

Function with two periods
Actually I think the OP was referring to one dimentional functions something like y=sin(x) + sin(Pi x) for example.
Rather than "having two period" (as far as I can see) such a function as aperiodic. 


#5
Jan2207, 09:20 PM

Sci Advisor
HW Helper
P: 2,537

There are also fractal functions. For example, consider the characteristic function of the rationals (f(x)=1 if x is rational, and 0 otherwise) which has a periods equal to any rational number. Or, if you prefer nonrational ratios, you could go with the characteristic function of the algebraic numbers (restricted to the real line).



#6
Jan2307, 02:41 AM

Emeritus
Sci Advisor
PF Gold
P: 4,500




#7
Jan2307, 02:45 AM

Sci Advisor
HW Helper
P: 9,396

Did you not read Nate's post?
It is elementary to show that any continuous periodic function on R with no smallest period is constant, and that anything with a smallest period has periods that are integer mutliples of this smallest period. However, it is easy to find noncontinuous functions on R that have infinitely many periods that are not integer multiples of each other and are not constant. Just pick the indicator function on any additive (divisible) subgroup of R, such as Q. 


#8
Jan2307, 07:04 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,533

Another example if f(x) is a noncontinuous function satisfying f(x+ y)= f(x)+ f(y), then it has every positive rational number as period.



#9
Nov1707, 03:00 PM

Emeritus
Sci Advisor
PF Gold
P: 4,500

Sorry to bump this, but I was looking at the problem again and was wondering if there was an official name for functions like the indicator function of an additive subgroup (or particularly a dense one) of R. I tried searching for stuff on fractal functions, but not surprisingly got a ton of stuff about Mandelbrot and Julia sets, then remembered about this post I made in times of Yore
Thanks for the help 


Register to reply 
Related Discussions  
Synchronization of periods  Biology  3  
Law of periods  Introductory Physics Homework  2  
Periods of continuous functions  Calculus  4  
Vibration periods  Introductory Physics Homework  7  
Comets and orbital periods and such  Introductory Physics Homework  1 