Can Fourier Analysis Represent Any Function Using Sin and Cos?

[LLT71]

has Fourier used sin(x) and cos(x) in his series because "there must be such interval [a,b] where integral of "some function"*sin(x) on that interval will be zero?" so based on that he concluded that any function can be represented by infinite sum of sin(x) and cos(x) cause they are "orthogonal" to any function.

let me recall my last thread on "orthogonality" as well to see if I got it right or missed the whole point:
https://www.physicsforums.com/threads/orthogonality-of-functions.891717/#post-5611182

extra question: is there possibility that you can find such "a" and "b" where ∫f(x)dx from a to b = zero or at least "try" using some "equation" or so?

thanks!

 

[BvU]

any function can be represented by infinite sum of sin(x) and cos(x) cause they are "orthogonal" to any function

That's not the case and that's not the idea.
You know a little about music ? There are tones and there are overtones (harmonics).
Let's keep it simple and start with periodic functions: functions for which $f(x+T)=f(x)$. Such $f$ can be written as $$ f(x) = \sum A_n\sin\left ({2\pi nx \over T}\right ) + B_n\cos\left ({2\pi nx \over T}\right )$$precisely because the functions $sin\left ({2\pi nx \over T}\right )$ and $cos\left ({2\pi nx \over T}\right )$ are orthogonal (wrt each other -- NOT wrt any function). See here what that means.

(If they were orthogonal to any function, all the $A_n$ and $B_n$ would be zero and you wouldn't get anywhere with your summation )

 

[LLT71]

See here what that means.

sorry for late reply (was away from home). your link doesn't work somehow. ok let's reformulate my question: out of all orthogonal functions why did he particularly choosed sin(x) and cos(x)?

 

[micromass]

Because Fourier was inspired by the wave equation.

 

[BvU]

sorry for late reply (was away from home). your link doesn't work somehow. ok let's reformulate my question: out of all orthogonal functions why did he particularly choosed sin(x) and cos(x)?

Harmonics is google search for "harmonic analysis in Fourier series" (google forced a .nl instead of .com in there, perhaps that's the reason it doesn't work).
The other was a paste error and I can't find it back. Crux was that $$\int \sin 2\pi nx \sin 2\pi mx =\delta_{nm}$$ (with some normalization) etc, as here or in point 6 here

 

[LLT71]

Harmonics is google search for "harmonic analysis in Fourier series" (google forced a .nl instead of .com in there, perhaps that's the reason it doesn't work).
The other was a paste error and I can't find it back. Crux was that $$\int \sin 2\pi nx \sin 2\pi mx =\delta_{nm}$$ (with some normalization) etc, as here or in point 6 here

thank you!

 

[annarosy]

the problem is difficult!

 

[BvU]

Hello Annarosy,

It is and it is not. But Fourier analysis definitely is a very powerful tool in science, so it's worth investing an effort.

Because Fourier was inspired by the wave equation

Check out (our google one that suits your tastes better) e.g. here or http://www.math.harvard.edu/archive/21b_fall_07/handouts/heat_and_wave_equation.pdf