Intrinsic Supermoothness

TL;DR

This paper reveals that supersmoothness, the enhanced smoothness observed at corners when gluing functions, is a fundamental property related to curve singularities and is not exclusive to polynomial-based splines.

Contribution

The authors demonstrate that supersmoothness arises from geometric properties of curves and extends beyond polynomial functions, providing a new understanding of smoothness phenomena.

Findings

Supersmoothness occurs at corners of curves, ensuring differentiability.

The phenomenon characterizes non-smooth curves in higher derivatives.

Supersmoothness is independent of polynomial properties.

Abstract

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.