Engineering Analysis/Arbitrary Basis Expansion

The classical Fourier series uses the following basis:

$\phi (x)={1,\sin(n\pi x),\cos(n\pi x)},n=1,2,...$

However, we can generalize this concept to extend to any orthogonal basis set from the L₂ space.

We can say that if we have our orthogonal basis set that is composed of an infinite set of arbitrary, orthogonal L₂ functions:

$\phi ={\phi _{1},\phi _{2},\cdots ,}$

We can define any L₂ function f(x) in terms of this basis set:

[Generalized Fourier Series]

$f(x)=\sum _{n=1}^{\infty }a_{n}\phi _{n}(x)$

Using the method from the previous chapter, we can solve for the coefficients as follows:

[Generalized Fourier Coefficient]

$a_{n}={\frac {\langle f(x),\phi _{n}(x)\rangle }{\langle \phi _{n}(x),\phi _{n}(x)\rangle }}$