hamsterman said:
A book I'm reading says that the set of continuous functions is an Euclidean space with scalar product defined as <f,g> = \int\limits_a^bfg and then defines Fourier series as \sum\limits_{i\in N}c_ie_i where c_i = <f, e_i> and e_i is some base of the vector space of continuous functions.
What I want to know is why that scalar product function was chosen. Would any function that has the properties of scalar multiplication do? Are there any other possible definitions and do they change some properties of the series?
Your Fourier series makes sense for other inner products as well. A general inner product on a set H is any function
<\cdot,\cdot><img src="/styles/physicsforums/xenforo/smilies/arghh.png" class="smilie" loading="lazy" alt=":H" title="Gah! :H" data-shortname=":H" />\times H\rightarrow \mathbb{K}
(with \mathbb{K}=\mathbb{R} or \mathbb{K}=\mathbb{C}
that satisfies
1) <x,x>\geq 0 for all x in H.
2) <\alpha x+\beta y,z>=\alpha <x,z>+\beta <y,z> for all x,y,z in H and 3) \alpha,\beta\in \mathbb{K}.
3) <x,y>=\overline{<y,x>} for all x,y in H (if \mathbb{K}=\mathbb{R} then this complex conjugate can be dropped of course)
This is a general inner product. There are many examples of this concept. Two important ones are:
1) On \mathbb{K}^n, we can define
<(x_1,...,x_n),(y_1,...,y_n)>=x_1\overline{y_1}+...+x_n\overline{y_n}
2) On a compact set \Omega\subseteq \mathbb{K}^n, we can let H be the continuous functions from \Omega to \mathbb{K}. An inner product can be given by
<f,g>=\int_\Omega f(x)\overline{g(x)}dx
These are the two most important examples of inner products.
But for any inner product, we can investigate the concept of Fourier series. That is, we usually have elements e_0,e_1,...\in H such that <e_n,e_n>=1 and <e_n,e_m>=0 for n and m distinct. The Fourier series of an element x is then defined as
\sum_n <x,e_n>e_n.
This is a formal series: we don't care about convergence. However, we can prove that it actually does converges for a certain norm.
In the example of \mathbb{K}^n above (let's take n=3 for convenience), we can take e_0=(1,0,0),~e_1=(0,1,0),~e_2=(0,0,1). Then the Fourier series of an element (x_1,x_2,x_3) is simply
x_1e_1+x_2e_2+x_3e_3
In this case, it equals (x_1,x_2,x_3). This is not always the case. To always have this, we want the e_n to be "complete".
Another thing I find odd is, why do I see differences on the Internet? Only trigonometric Fourier series are talked about. Does general decomposition of a function into a series go by some other name?
The Fourier series you see on the internet are a very special case of the above concept. For them, we take H = the continuous functions from [0,2\pi[ to \mathbb{K} (note that [0,2\pi[ is not compact, which is a technical difficulty). The inner product is that of (2).
We take the functions
e_0=\frac{1}{\sqrt{2\pi}}
e_{2n}=\frac{1}{\sqrt{\pi}} \sin(nx) for n>0
e_{2n+1}=\frac{1}{\sqrt{\pi}} \cos(nx) for n>0
(This is not standard notation)
In that case, the Fourier series of an arbitrary function f can be written as
\frac{a_0}{2}+\sum_{n=1}^{+\infty} a_n \cos(nx) + \sum_{n=1}^{+\infty} b_n \sin(nx)
for certain a_n,b_n. (ignore the division by 2 in a_0, this is to make certain formula's easier).
This is the Fourier series that is widely known as Fourier series. But it's just a special case of the general concept.
