Engineering Analysis/Arbitrary Basis Expansion

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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}