interpolare

Interpolare is the Italian verb meaning to interpolate. In mathematics and data analysis, interpolation is the process of estimating unknown values within the range of a discrete set of known data points. It aims to construct a function that matches the known data exactly at the given points, enabling predictions or representations of a continuous phenomenon. By contrast, extrapolation estimates values outside the observed range.

Formal setup: Suppose you have nodes x0 < x1 < ... < xn with corresponding values y0, y1, ..., yn. There exists a unique polynomial P of degree at most n such that P(xi) = yi for all i. Common representations include the Lagrange form P(x) = sum_{i=0}^n yi Li(x), where Li(x) = product_{j≠i} (x - xj)/(xi - xj). Newton’s divided-difference form and the barycentric form are numerically popular; all yield the same interpolant when data are exact.

Methods: Linear interpolation uses two points to estimate values between them. Polynomial interpolation uses a single polynomial; spline interpolation builds piecewise polynomials—often cubic—joined with continuity constraints. High-degree polynomial interpolation on equispaced nodes can suffer the Runge phenomenon; splines and other local methods mitigate this.

Applications: Interpolation is used in numerical analysis, data fitting, image and signal processing, computer graphics, and geostatistics for estimating missing measurements or resampling data. It is distinct from extrapolation, which estimates outside the known range.