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emyt
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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
thanks for sharing
HallsofIvy said:With what period? I cannot imagine how to do that!
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!
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.
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).
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.
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.
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.
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.