Suppose we want to interpolate an infinite number of points on the Cartesian plane using a continuous and differentiable function . How can this be done?
Given points on the Cartesian plane, the set of points can be interpolated using a polynomial of at least degree . Given an infinite number of points to interpolate, we need an infinite polynomial:
where is within the radius of convergence.
Solving for each constant term expands the original function into the infinite polynomial:
Check out my other notes at Proof, Disproof, and Derivation