basis function

Concept

Interpolating function $f$ can be define as linear combination of basis functions

$$f(t)=\sum _{j=1}^n{x}_j{\varphi }_j(t)$$

Fourier basis

comprising sines and coses. For example a set of basis functions with $n=2k$ in $[0,2\pi ]$:

$$\left[\begin{array}{cccc}{\varphi }_1(t)& {\varphi }_2(t)& \cdots & {\varphi }_n(t)\end{array}\right]$$ $$\left[\begin{array}{cccccccc}\cos (0)& \cos (1t)& \cdots & \cos (kt)& \sin (t)& \cos (2t)& \cdots & \cos ((k-1)t)\end{array}\right]$$

Monomial basis (bad)

$${\varphi }_j(t)=t^{j-1}$$

specifically for this one, we need to use Vandermonde matrix to solve. It require $O(n^3)$ to solve. Considering the bad performance and unstable, we should use other instead.

Newton Basis

$${\varphi }_j(x)=\prod _{i=0}^{j-1}(x-x_i)$$