# F(x) = x as a sum of periodic functions?

1. Oct 18, 2009

### emyt

someone told me that there's a proof that says f(x) = x can be expressed as a sum of two periodic functions.. does anybody know this?

thanks for sharing

2. Oct 18, 2009

### HallsofIvy

With what period? I cannot imagine how to do that!

3. Oct 18, 2009

### emyt

I'm not sure, someone just told me.. I don't really see how that would work, but who knows?

Last edited: Oct 18, 2009
4. Oct 18, 2009

### arildno

To express it as the sum of INFINITELY many periodic functions on some interval is trivial, however..

With two? Give me some more, please!

5. Oct 18, 2009

### emyt

I'm not quite sure how a function as the sum of an infinite amount of periodic functions can be f(x) = x, could you explain please?

thank you

Last edited: Oct 18, 2009
6. Oct 18, 2009

### lstellyl

so i'm gonna go ahead and take a guess that your friend was referring to a fourier series.

as arildno has pointed out, this is a sum of an infinite number of periodic functions, not just two. (ie, it would be an infinite sum of sines, each with a different period and amplitude)

you can theoretically do the same with any periodic function, though fourier was kind enough to show us how to calculate the coefficients and periods to describe any function we wish as just a sum of sines and cosines, so that is what is most often used.

there will be a lot of stuff on the internet that goes into detail on proving this, as well as showing you precisely what is going on...

also, so a function is periodic with an interval P if f(x)=f(x+P)

7. Oct 18, 2009

### arithmetix

Fourier got a hard time from his contemporaries, who could not accept that a square wave may be expressed as the infinite sum of a set of sin functions. Nowadays we are shown at an early stage of our (electronic communications) studies that Fourier was right. You could do worse than consult a communications text in your attempt to come to grips with this non-intuitive notion.
For the sake of decency I include the details of an infinite series which describes a square wave:
sq(t)=sin(t) + (1/3)sin(3t) + (1/5)sin(5t) + (1/7)sin(7t) + ....
Try plotting this series, successively using more and more terms, and you will see the square wave taking shape as you go.

8. Oct 18, 2009

### lstellyl

this is also a wonderful java applet that shows the effect quite well:

i remember when i was first learning about fourier series... took me a while to be comfortable with the idea of it all

9. Oct 18, 2009

### emyt

thanks, I know nothing about fourier series.. how does this relate to f(x) = x as a combination periodic functions?

from my own observations, I have seen that the sum of periodic functions can either be periodic or not quite periodic but "close enough". So, if you took enough of these periodic functions that created an "almost" periodic function ( I've heard this term actually defined, but I don't know the definition so I'm using it in a non-rigourous sense), you could get something that becomes not periodic at all. any thoughts on this? could you some how get f(x) = x in this way?

10. Oct 18, 2009

### Hurkyl

Staff Emeritus
Can you write down a sum of one or more periodic functions that equals f on the interval [-1,1]?
Now, can you add to that a sum of one or more periodic functions that equals f on the interval [-2,2]?
Now, can you add to that a sum of one or more periodic functions that equals f on the interval [-4,4]?

11. Oct 19, 2009