Intrinsic Supermoothness
This work clarifies the geometric origin of supersmoothness for mathematicians studying smooth functions and splines, but is incremental as it uses standard calculus tools.
The paper identifies and characterizes a phenomenon called 'supersmoothness' in continuously glued smooth functions along non-smooth curves, showing it is a general property of smooth functions rather than polynomial splines, and generalizes it to higher derivatives.
The phenomenon, known as "supersmoothness" was first observed for bivariate splines and attributed to the polynomial nature of splines. Using only standard tools from multivatiate calculus, we show that if we continuously glue two smooth functions along a curve with a "corner", the resulting continuous function must be differentiable at the corner, as if to compensate for the singularity of the curve. Moreover, locally, this property, we call supersmoothness, characterizes non-smooth curves. We also generalize this phenomenon to higher order derivatives. In particular, this shows that supersmoothness has little to do with properties of polynomials.