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Spline

Spline:

A spline is a continuous function Math image which coincides with a polynomial on every subinterval Math image of the whole interval Math image on which Math image 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 Math image of the polynomial is the same. Splines are often named after the order Math image of the spline, e.g. cubic splines correspond to Math image .

An essential feature of splines is that function Math image is continuous – i.e. has no breaks on the boundaries between two adjacent intervals. Besides the continuity of the function Math image itself, for many types of splines the first derivatives of Math image are also continuous. (For example, for cubic splines the first derivative Math image is continuous too, while derivatives of higher order, e.g. Math image 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 Math image of a single variable Math image ), there are also two-dimensional splines, which are functions Math image of two variables, say, Math image and Math image . Two-dimensional splines are used, for example, to interpolate spatial fields from results of their measurements in a finite set of Math image points Math image .

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