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

In summary, there is a proof that states f(x) = x can be expressed as a sum of infinitely many periodic functions on some interval. This is known as a Fourier series, discovered by Fourier, and it allows any periodic function to be described as a sum of sine and cosine functions. It was initially met with skepticism but is now widely accepted in the field of electronic communications. It is possible to represent a non-periodic function with a sum of periodic functions, but it may not be a perfect representation. This concept can also be applied to other intervals, as shown by the example of finding sums that equal f(x) on different intervals.
  • #1
emyt
217
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
 
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  • #2
With what period? I cannot imagine how to do that!
 
  • #3
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?
 
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  • #4
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
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
 
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  • #6
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)
 
  • #7
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
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
 
  • #9
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]?
 

1. What is the meaning of "F(x) = x as a sum of periodic functions"?

The function F(x) = x as a sum of periodic functions means that the function F(x) can be expressed as the sum of multiple periodic functions. This means that the function repeats itself after a certain interval, and the sum of these repeating functions results in the original function F(x).

2. What are periodic functions?

Periodic functions are functions that repeat themselves after a certain interval. This interval is called the period and is represented by the variable T. Examples of periodic functions include sine and cosine functions.

3. How can F(x) be expressed as a sum of periodic functions?

F(x) can be expressed as a sum of periodic functions by identifying the period of the function and finding a combination of periodic functions that when added together, result in the original function. This can be done by using Fourier series or other mathematical techniques.

4. What are the applications of expressing F(x) as a sum of periodic functions?

Expressing F(x) as a sum of periodic functions is useful in various fields such as signal processing, image processing, and data compression. It allows us to represent complex functions in a simpler form, making it easier to analyze and manipulate them.

5. Are all functions able to be expressed as a sum of periodic functions?

No, not all functions can be expressed as a sum of periodic functions. Only functions that are periodic themselves can be expressed in this way. Functions that do not have a repeating pattern cannot be expressed as a sum of periodic functions.

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