Interpolation

A type of estimation. A method of constructing new data points based on the range of a discrete set of known data points. It is the process of deriving a simple function from a set of discrete data points so that the function passes through all the given data points and can be used to estimate data points in-between the given ones.

Weierstrass Approximation Theorem

Suppose $f$ is continuous on $[a,b]$. Then $\forall \epsilon \gt 0,\;\exists P(x)$ such that $\forall x \in [a,b]$:

$$\left\lvert f(x) - P(x) \right\rvert < \epsilon$$

Note

For $n + 1$ data points, there is one and only one polynomial of order $n$ that passes through all the points.