 Spline approximation

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.