Function with two periods

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In summary, it was discussed that a function with two periods a and b does not necessarily have to be constant, as long as a and b are rational multiples of each other. It was also mentioned that functions with infinitely many periods that are not integer multiples of each other can exist, such as the indicator function of an additive subgroup of R. These types of functions may fall under the category of fractal functions, but further research is needed to determine an official name.
  • #1
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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 b-a < b, and either a < b-a or a > b-a. 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 p-e is a period, and e-(p-e) = 2e-p is a period. 2e-p - (p-e) = 3e-2p is a period, and so is 3e -2p + (2e-p) 5e-3p. 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 ki*a - mi*b, where, I believe, ki/mi approaches b/a.

Thoughts?
 
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  • #3
Ahh I am not too sure about anything, but i was wondering how a function could have to periods, and then I realized the most obvious answer of extending in more diemensions :)
 
  • #4
Actually I think the OP was referring to one dimensional 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
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 non-rational ratios, you could go with the characteristic function of the algebraic numbers (restricted to the real line).
 
  • #6
uart said:
Actually I think the OP was referring to one dimensional 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.

I was referring to one dimensional functions. I'm fairly certain that a function can't have two periods that aren't multiples of each other, but am wondering how to prove that
 
  • #7
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.
 
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  • #8
Another example- if f(x) is a non-continuous function satisfying f(x+ y)= f(x)+ f(y), then it has every positive rational number as period.
 
  • #9
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 sub-group (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
 

1. What is a "Function with two periods"?

A function with two periods is a mathematical concept in which a repeating pattern occurs twice within one period of the function. This means that the function will have two peaks or two valleys within one cycle.

2. How is a "Function with two periods" different from a regular function?

A regular function has only one period, meaning that the pattern repeats itself only once within one cycle. A function with two periods has two distinct patterns that repeat within one cycle, making it more complex than a regular function.

3. What is the formula for a "Function with two periods"?

The formula for a function with two periods is f(x) = A sin(Bx) + C cos(Dx), where A, B, C, and D are constants that determine the amplitude, frequency, and phase shift of the function.

4. How can I graph a "Function with two periods"?

To graph a function with two periods, you can plot points for one full cycle of the function and then use the symmetry of the function to plot the remaining points. Alternatively, you can use a graphing calculator or software to plot the function accurately.

5. What are some real-life examples of a "Function with two periods"?

Some real-life examples of functions with two periods include sound waves, ocean tides, and pendulum swings. These phenomena exhibit two distinct patterns that repeat within one cycle, making them good examples of functions with two periods.

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