= 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 $|f''|\leq M$ on each cell of width $h$, the interpolation remainder $f(x)-p(x)=f''(\xi)(x-a)(x-b)/2$ gives $\|f-p\|_\infty\leq Mh^2/8$. 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>.
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