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

AI Thread Summary
The discussion centers on whether the function f(x) = x can be expressed as a sum of two periodic functions. Participants express skepticism about this claim, noting that while it is trivial to represent f(x) as a sum of infinitely many periodic functions, the challenge lies in doing so with just two. The concept of Fourier series is introduced, which allows for the representation of functions as sums of sine and cosine functions, but emphasizes that this involves infinite terms rather than a finite sum. The conversation also touches on the idea that sums of periodic functions can yield non-periodic results, raising questions about how f(x) = x could be approximated in this manner. Overall, the thread highlights the complexities of combining periodic functions to achieve non-periodic outcomes.
emyt
Messages
217
Reaction score
0
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
 
Mathematics news on Phys.org
With what period? I cannot imagine how to do that!
 
HallsofIvy said:
With what period? I cannot imagine how to do that!

I'm not sure, someone just told me.. I don't really see how that would work, but who knows?
 
Last edited:
To express it as the sum of INFINITELY many periodic functions on some interval is trivial, however..

With two? Give me some more, please!
 
arildno said:
To express it as the sum of INFINITELY many periodic functions on some interval is trivial, however..

With two? Give me some more, please!

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:
so I'm going to 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)
 
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.
 
this is also a wonderful java applet that shows the effect quite well:
http://www.falstad.com/fourier/

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

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
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]?
 

Similar threads

A Can You Prove the Series Is Periodic?
Replies
33
Views
3K
A Measure of non-periodicity of almost periodic functions
Replies
1
Views
2K
I Periodic Functions: Meaning of 1-Periodicity
Replies
2
Views
1K
I Mean of the derivative of a periodic function
Replies
6
Views
2K
MHB ACT.trig.01 What is the period of the function csc 4x
Replies
2
Views
2K
I My attempt to understand horizontal transformations of functions
Replies
6
Views
2K
I Period of a Sine Wave: Understand How to Measure in Radians
Replies
6
Views
2K
MHB Graphing Trig Function: Amplitude 4, Period (2\pi/3), Range [-4,4]
Replies
1
Views
2K
Sum of Related Periodic Functions
Replies
5
Views
2K
B Looking for a specific periodic function
Replies
4
Views
2K

Hot Threads

Recent Insights

Back
Top