interpolates

Interpolates is the third-person singular present tense of interpolate, a verb meaning to estimate a value within the range of a discrete set of known data points. In mathematics, statistics, and data analysis, interpolation is the process of constructing a function that passes through or closely approximates the given data so that values at unsampled points can be inferred. Interpolation is distinct from extrapolation, which estimates values outside the known data range.

Common interpolation methods include linear interpolation, which connects adjacent points with straight segments; polynomial interpolation, which fits a polynomial that exactly passes through the data (for example, using Lagrange or Newton forms); and spline interpolation, which uses piecewise low-degree polynomials to produce a smooth curve. Other approaches include rational, piecewise polynomial, and multivariate interpolation for higher dimensions. Each method balances simplicity, smoothness, and fidelity, and some can suffer from issues such as Runge’s phenomenon with high-degree polynomials.

In practical use, interpolation is employed to resample data, fill in missing values, approximate functions, or upsample signals and images. In signal processing, techniques such as sinc or polynomial-based interpolation estimate samples at nonuniform times; in computer graphics, bilinear or bicubic interpolation scales images; in geographic information systems, interpolation estimates values on grids from scattered observations.

Etymology and usage: interpolate derives from Latin interpolare, meaning “to insert.” Interpolation schemes are chosen based on data behavior, boundary conditions, and the required accuracy.