Interpolation

A method of constructing new data points within the range of a discrete set of known data points. The constructed function passes through all given points.

Weierstrass Approximation Theorem

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

Types

Linear Interpolation

Fits a straight line between 2 adjacent data points $(x_0, y_0)$ and $(x_1, y_1)$:

Simple but only accurate when the underlying function is nearly linear between nodes.

Polynomial Interpolation

Fits a single polynomial of degree $n$ through $n + 1$ data points. Methods include Lagrange and Newton’s divided differences.

High-degree polynomials can oscillate wildly near the endpoints (Runge’s phenomenon).

Spline Interpolation

Fits piecewise polynomials between consecutive data points, joined smoothly at knots. Cubic splines are most common: each piece is degree 3 and the join points satisfy $C^2$ continuity (matching function value, first derivative, and second derivative).

Avoids Runge’s phenomenon while remaining smooth.