Engineering Analysis/Arbitrary Basis Expansion

The classical Fourier series uses the following basis:

$\varphi (x)=1,\sin (n\pi x),\cos (n\pi x),n=1,2,...$

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:

$\varphi ={\varphi }_1,{\varphi }_2,\cdots ,$

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

$f(x)=\sum _{n=1}^{\mathrm{\infty }}{a}_n{\varphi }_n(x)$

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

$a_n=\frac{⟨f(x),{\varphi }_n(x)⟩}{⟨{\varphi }_n(x),{\varphi }_n(x)⟩}$