A basis function is an element of a set of functions that span a function space.
Basis functions are spanning sets of functions. These include sums of sine waves to match any waveform, radial basis functions used for artificial neural networks, basis vectors of a linear algebra, and $\Sigma_0^{n-1} t^i$ to span a polynomial of degree $n-1$. For the latter, $\Sigma t^i$ is called a cardinal or standard basis because the coefficients of $t^i$ are all 1's.
