Expansions and Basis Functions

The above power series express a function such as $\sin(x)$ in terms of an infinite series of power functions (like Axⁿ) all added together. More generally, almost any function f(x)can be represented perfectly as a linear combination of many other types of functions besides power functions:

$$f(x)= \sum_{k}a_{k}\Psi_{k}(x)$$ (7)

where the chosen $\Psi_{k}(x)$ are called expansion basis functions. For example, in the case of Fourier expansions in one dimension, the expansion basis functions are the complex exponentials:

$$\Psi_{k}(x) = \exp(i\mu_{k} x)$$ (8)

``Finding the representation of some function in a chosen basis" means finding the set of coefficients a_k which, when multiplied by their corresponding basis functions $\Psi_{k}(x)$ and the resulting linear combination of basis functions are summed together, will exactly reproduce the original function f(x) as per Eqt. (8).

This is a very powerful tool, because it allows one to choose some universal set of functions in terms of which all other (well-behaved) functions can be represented just as a set of coefficients! In the case of systems analysis, a major benefit of doing this is that knowledge about how members of the chosen universal set of basis functions behave in the system gives one omniscient knowledge about how any possible input function will be treated by the system.