Interpolatable

Interpolatable describes the property of a dataset, function, or model that allows interpolation—estimating values at points between known samples. In mathematics, an interpolation problem seeks a function f from a chosen class such that f(x_i) = y_i for all i. For a finite set of distinct x_i, there often exists an interpolant of sufficient flexibility, for example a polynomial of degree at most n-1 passing through n points.

Common interpolation methods include polynomial interpolation (Lagrange, Newton forms), spline interpolation (such as cubic splines) which piece together smooth polynomials, and multivariate techniques such as tensor-product splines, radial basis functions, and kriging for scattered data. The choice of function class influences interpolation behavior and numerical stability.

Interpolation is distinct from extrapolation; interpolants may behave poorly outside the known domain, and high-degree polynomials can exhibit Runge’s phenomenon or become numerically unstable. With noisy data, exact interpolation can overfit, so smoothing or regularized approaches—such as smoothing splines or regression-based methods—are often preferred to obtain a usable interpolant.

Applications span data visualization, numerical simulation, geostatistics, and computer graphics, as well as signal processing and scientific computing. In machine learning and related fields, the idea of interpolatability also appears in discussions of latent spaces or generative models, where interpolating between samples should yield plausible intermediate outputs.