Greetings
In this webpage, I am going to teach some math, some really cool math. So, are you ready for \(\LaTeX\)?
Definition
We define the L2 inner product of two elements from the function space as the following:
$$<f,g>_{L2} = \int_{a}^b fg\;dx \quad \forall f,g \in \mathcal{F}(\mathbb{R},\mathbb{R})$$ Where f and g are real function that has domain \([a,b]\) to \(\mathbb{R}\) and \(b>a\).
Orthogonal set
It's trivial that the real function space is a linear subspace, the proof is omitted here. A linear subspace will alaways has a orthogonal basis, for example, please consider the following set: $$ \lbrace cos\left( \frac{2\pi i x}{b-a} \right) \rbrace_{i=0}^{\infty} \cup \lbrace sin\left( \frac{2\pi i x}{b-a} \right) \rbrace_{i=1}^{\infty} $$ This is the set that makes all the magic because it's an orthoganal set under the real function space. And if you are smart, or I should say, knowladgeable, you will find out that this set is the basis for the fourier series.
However, this is not the end of the story, there are infinitely many orthogonal set presents in the L2 function space, in fact, if I were to scale the the element in the previous set by a non zero factor, we will have another ortho set, because, well, the set spans a linear subspace. So I would like to prahse it as:
There are as many ortho basis as you want.-Hongdad Li
Ortho Basis Varification
Assume that we have a countable set of functions and it formed an ortho basis: \(\lbrace f_i \rbrace\) then it means that: $$ < f_i, f_j > =0 \quad \forall j\neq i \text{ and } < f_i, f_j > \neq0 \quad \forall j= i $$ Well, in that case, the above ortho basis can be tested by proving the following statements: $$ \begin{align} \int_a^b \left( cos\left( \frac{2\pi i x}{b-a} \right) + sin\left( \frac{2\pi i x}{b-a} \right) \right) \left( cos\left( \frac{2\pi j x}{b-a} \right) + sin\left( \frac{2\pi j x}{b-a} \right) \right) dx & =0 \quad \forall i\neq j \wedge i,j> 0 \\ \int_a^b\left( cos\left( \frac{2\pi i x}{b-a} \right) + sin\left( \frac{2\pi i x}{b-a} \right) \right) dx & =0 \quad \forall i > 0 \end{align} $$ Some of you might notice that if we were to get rid of the sin and cos with Euler formula it might make thing easier, and it's true, in fact, that is where the imaginary exponent in other definitions of fourier series come from.
Other Ortho Set
At this stage you might wonder what are some of the other cool ortho set for the function space? Well, in that case I do have one other ortho set in mind, that is the Legendre Polynomial which is really fascinating because it's constructed through Gram Schmit process on the countable set: \(\{x^n\}_{n=0}^{\infty}\).