A spline is a continuous function which coincides with a polynomial on every subinterval of the whole interval on which is defined. In other words, splines are functions which are piecewise polynomial. The coefficients of the polynomial differs from interval to interval, but the order of the polynomial is the same. Splines are often named after the order of the spline, e.g. cubic splines correspond to .
An essential feature of splines is that function is continuous - i.e. has no breaks on the boundaries between two adjacent intervals. Besides the continuity of the function itself, for many types of splines the first derivatives of are also continuous. (For example, for cubic splines the first derivative is continuous too, while derivatives of higher order, e.g. are not continuous). This property makes splines look as rather smooth functions.
Splines are widely used for interpolation and approximation of data sampled at a discrete set of points - e.g. for time series interpolation.
Besides one-dimensional splines (i.e. functions of a single variable ), there are also two-dimensional splines, which are functions of two variables, say, and . Two-dimensional splines are used, for example, to interpolate spatial fields from results of their measurements in a finite set of points .