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I have been studied classical mechanics and quantum mechanics a little and after that,

I got a kind of feeling "I have to study Fourier analysis(FA) again".

So I have been studying FA again.

Here're my question.

1. From my viewpoint, FA is a method of expanding functions into several fraction of wave function using orthogonality. how do I put this? like using a lot of shadows of some physics phenomenon on each plains.?(For FA, wave function) so from one plain, one side of phenomenon will be seen and other sides will not be seen.

Is it right?

2. And about FA solution. For simple oscillation problem with two variables(horizontal axis "x" and time variable t) , initial displacement function called f(x) and initial velocity 0(m/s or any kind of velocity demension), assume that I got a simple solution with variable x, t which have undetermined constant.

for your clear understanding of situation, like this.

solution U=XT=(Acosx + Bsinx)*(Ccost + Dsint)

from here, there's a problem.

with two initial condition, FA textbook says,

for initial velocity condition, you have to just get a derivate of function U and set t=0.

but for initial velocity condition, you have to get a Fourier expansion of U and set t=0.

What's the reason of that? if textbook said about only one for derivate or Fourier expansion, I would have no question about that. but why?

I'm wondering how I can interpret that solution physically and mathematically.

If someone can answer this without math, I would appreciate it. because that is more beautiful.

regards and sorry for my poor English.

regards.