# Engineering Analysis/Arbitrary Basis Expansion

The classical Fourier series uses the following basis:

However, we can generalize this concept to extend to any orthogonal basis set from the L_{2} space.

We can say that if we have our orthogonal basis set that is composed of an infinite set of arbitrary, orthogonal L_{2} functions:

We can define any L_{2} function f(x) in terms of this basis set:

[Generalized Fourier Series]

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

[Generalized Fourier Coefficient]