General Fourier Series
J.B. Fourier 1768-1830
Given a function f(t)
defined over an interval from t1 to t2, we can write the Generalized Fourier Series:
where the {Xn(t)} is a complete orthogonal bases . This means two things:
· any function f can be expressed in terms of these bases functions
· the Xn(t) are orthogonal to one another
Xn(t)·Xm(t)
= 0 = 
if m ≠ n
The Cn can then be found as:
* denotes complex
conjugate if the bases involves complex functions
where 
(in some cases a weight function may be involved in the
inner product)
The En are referred to as energy coefficients to electrical engineers. The next section on the trigonometric Fourier Series discusses this interpretation further.
The actual orthogonal set, {Xn(t)}, depends on the situation one is studying. Examples of these are Legendre polynomials, Bessel functions and Hermite polynomials, as well as the classic example of trigonometric functions. The overall study of differential equations which generate orthogonal solutions is done in Sturm-Liouville theory, a branch of Boundary Value Problems.
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