I have recently started looking at the topoic of Fourier series. Consider the space of square integrable functions $L_{2}[0,2\pi]$. Where we define the inner product as $(f,g):= \int_{0}^{2\pi}fg dx$ and the corresponding norm $\Vert f \Vert = \sqrt{(f,f)}$. The fourier series of a function $f(x)$ in this space is defined as $$f(x) = a_{0} + \sum_{n=1}^{\infty}a_{n}cos(nx) + \sum_{n=1}^{\infty}b_{n}sin(nx)$$
Alternativley it says we can consider the set $$\{ \frac{1}{\sqrt{2\pi}},\frac{1}{\sqrt{\pi}}cos(nx),\frac{1}{\sqrt{\pi}}sin(nx),...\} ~~~~n=1,2,...$$ Which is orthonormal. Then for any function in $L_{2}$ the series converges in the $L_{2}$ norm.
Questions:
Why is the fourier expansion unique to square integrable functions $L_{2}[0,2\pi]$, why is this space chosen?
Can the orthonormal set defined above $\{ \frac{1}{\sqrt{2\pi}},\frac{1}{\sqrt{\pi}}cos(nx),\frac{1}{\sqrt{\pi}}sin(nx),...\} ~~~~n=1,2,...$ be considered a Schaduer basis for $L_{2}[0,2\pi]$?
Thanks for any assistance, I am just starting to learn about Fourier series so any assistance is appreciated.
In the 1700's these functions came up in looking at the vibrations of a string, and someone noticed that one could isolate the coefficients of an expansion $$ f \sim \frac{1}{2}a_{0}+a_{1}\cos x+b_{1}\sin x+ a_{2}\cos 2x + b_{2}\sin 2x +\cdots $$ using an integral orthogonality condition: it was discovered that the integral of any function in $\{ 1,\cos x,\sin x,\cos 2x,\sin 2x,\cdots\}$ against any different such function would result in $0$ for the answer. In that way one could isolate the coefficients by multiplying the above expansion by one of these functions and integrating to isolate one coefficient. For example, $$ \int_{0}^{2\pi}f(x)\cos(mx)\,dx = a_{m}\int_{0}^{2\pi}\cos^{2}(mx)\,dx = \pi a_{m}. $$ Fourier conjectured that any function could be expanded in this way, whereas people before him believed that these constraints limited what functions could be expanded in this way.
Eventually, this was abstracted to an inner product $(\cdot,\cdot)$ on functions that looked like dot product on vectors in $\mathbb{R}^{n}$, where one defines $$ (f,g) = \int_{0}^{2\pi}f(x)g(x)\,dx. $$ This definition was immediately useful in isolating the Fourier coefficients. The only real qualification for a function to be considered was that $$ (f,f) = \int_{0}^{2\pi}f^{2}(x)\,dx < \infty. $$ That led naturally to $L^{2}[0,2\pi]$ defined as the space of square-integrable functions. For any two such function $f,g \in L^{2}[0,2\pi]$, this 'dot' product integral is absolutely convergent because $$ |fg| \le \frac{1}{2}f^{2}+\frac{1}{2}g^{2}, $$ a fact which follows from the simple inequality $(a-b)^{2} \ge 0$ or $a^{2}+b^{2} \ge 2ab$ for any positive $a$, $b$.
Parseval's equality (~1795) connects square integrable functions and square-summable sequences because of the mutual orthogonality of the functions: $$ \frac{1}{\pi}\int_{0}^{2\pi}f^{2}(x)\,dx = a_{0}^{2}+a_{1}^{2}+b_{1}^{2}+a_{2}^{2}+b_{2}^{2}+\cdots. $$ There is a perfect correspondence between $L^{2}[0,2\pi]$ and the infinite-dimensional generalization of Euclidean space where the vectors have coordinates $(a_{0},a_{1},b_{1},a_{2},b_{2},a_{3},b_{3},\cdots)$. The dot products correspond perfectly, too. If $$ f \sim (a_{0},a_{1},b_{1},a_{2},b_{2},\cdots) \\ g \sim (a_{0}',a_{1}',b_{1}',a_{2}',b_{2}',\cdots), $$ then $$ (f,g)=\frac{1}{\pi}\int_{0}^{2\pi} fg\,dx = a_{0}a_{0}'+a_{1}a_{1}'+b_{1}b_{1}'+a_{2}a_{2}'+\cdots $$ That correspondence was enough to validate the study of $L^{2}[0,2\pi]$ as an important space in its own right, and as the correct place to study the Fourier series. It took until about 1885 to realize that $$ \|f\| = (f,f)^{1/2} $$ was a way to define a norm for general function spaces, and could be useful for discussing convergence.
Schauder Basis: The functions you give form a complete orthonormal basis for $L^{2}[0,2\pi]$. If $\{e_{n}\}_{n=0}^{\infty}$ denotes the elements of your set in order, then $$ \lim_{N\rightarrow 0}\left\|f - \sum_{n=N}^{\infty}(f,e_{n})e_{n}\right\|_{L^{2}[0,2\pi]}=0 $$ holds for every $f \in L^{2}[0,2\pi]$. Using the usual definition for a Schauder basis, this qualifies $\{ e_{n}\}$ as a Schauder basis. This orthonormal basis is where all of the infinite-dimensional discussions started, and it is the prototype for orthonormal bases. Schauder bases are generalizations of such Hilbert Space bases to other types of spaces, especially Banach spaces. The $L^{2}[0,2\pi]$ convergence of orthogonal expansions is equivalent to Parseval's equality holding for all $f\in L^{2}$: $$ \|f\|^{2}_{L^{2}}=\sum_{n=0}^{\infty}|(f,e_{n})|^{2},\;\;\; f\in L^{2}[0,2\pi]. $$ Another equivalent is that the following holds for all $f,g\in L^{2}$: $$ (f,g)_{L^{2}} = \sum_{n=0}^{\infty}(f,e_{n})(e_{n},g). $$