- #1
Gianmarco
- 42
- 3
I'm wondering if anyone could give me the intuition behind Fourier series. In class we have approximated functions over the interval ##[-\pi,\pi]## using either ##1, sin(nx), cos(nx)## or ##e^{inx}##.
An example of an even function approximated could be:
##
f(x) = \frac {(1,f(x))}{||1||^{2}}*1 + \sum^{inf}_{n=1}\frac{(cos(nx), f(x))}{||cos(nx)||^{2}}*cos(nx)
##
where I've indicated the scalar product as (. , .) and the norm as || . ||.
From what I've understood, whenever computing the approximation using sin(nx) or cos(nx) the index of the sum starts from 1 and goes to infinity. Whenever dealing with ##e^{inx}## the index starts from - infinity and goes to infinity.
I think that when we compute the series approximation, we calculate the orthogonal projection of our original function on the infinite number of axis' given by ##1, sin(nx), cos(nx)## or ##e^{inx}##. Is this correct? And if it is, then why does the index start from 1 for sine and cosine whereas it's the whole integer set for the exponential base? Any help is much appreciated! :)
An example of an even function approximated could be:
##
f(x) = \frac {(1,f(x))}{||1||^{2}}*1 + \sum^{inf}_{n=1}\frac{(cos(nx), f(x))}{||cos(nx)||^{2}}*cos(nx)
##
where I've indicated the scalar product as (. , .) and the norm as || . ||.
From what I've understood, whenever computing the approximation using sin(nx) or cos(nx) the index of the sum starts from 1 and goes to infinity. Whenever dealing with ##e^{inx}## the index starts from - infinity and goes to infinity.
I think that when we compute the series approximation, we calculate the orthogonal projection of our original function on the infinite number of axis' given by ##1, sin(nx), cos(nx)## or ##e^{inx}##. Is this correct? And if it is, then why does the index start from 1 for sine and cosine whereas it's the whole integer set for the exponential base? Any help is much appreciated! :)