Quadratic spline ($q\in C^1,q{|}_{(x_i,x_{i+1})}\in {\mathcal{P}}_2$)

A quadratic spline with simple knots is a continuously differentiable function that is a polynomial of degree at most two between successive knots. Its first derivative is continuous and piecewise linear, while its second derivative is piecewise constant and can jump at knots. Constant tails impose vanishing first derivatives at the outermost knots. Such a spline arises from first-derivative roughness penalization when observations are interval averages, as in the quadratic smoothing spline for interval averages.