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A regression spline fits a regression function in a finite-dimensional spline approximation space with selected knots. Evaluating its basis at predictor values gives a design matrix, reducing fitting to regression on basis coefficients.

Spline approximation represents a function by low-degree polynomial pieces joined at specified knots. Smoothness conditions control the agreement of neighboring pieces. This permits local changes without fitting one high-degree polynomial across the entire domain.

The simplest example is piecewise-linear interpolation on a uniform grid. If $|f^{\prime \prime }|\le M$ on each cell of width $h$, the interpolation remainder $f(x)-p(x)=f^{\prime \prime }(\xi )(x-a)(x-b)/2$ gives $\Vert f-p{\Vert }_{\infty }\le M{h}^2/8$. More regular cubic splines impose derivative continuity as well. Spline quasi-interpolation uses local functionals instead of solving for a global interpolating fit, making locality useful in both analysis and computation.

A spline is a piecewise polynomial function whose polynomial pieces obey specified smoothness conditions at their joins, called spline knots. For example, a cubic spline is cubic on each interval and commonly has two continuous derivatives across interior knots. Spline approximation and spline interpolation use these finite-dimensional function spaces to represent data or approximate functions.