Fourier Series
If
- Dirichlet conditions are satisfied
then we can write
The above equation is called the complex fourier series. Given , we may determine by taking the inner product of with .
Let us assume a solution for of the form . Now we take the inner product of with .
If then,
If then,
We can simplify the above two conclusion into one equation.
So, we may conclude
Orthogonal Functions
The function and are orthogonal on if and only if .
The set of functions are orthonormal if and only if .
Linear Systems