A class of nowhere differentiable functions satisfying some concavity-type estimate

Abstract

We introduce and investigate a class $\mathcal {P}$ of continuous and periodic functions on $\mathbb {R}$. The class $\mathcal {P}$ is defined so that second-order central differences of a function satisfy some concavity-type estimate. Although this definition seems to be independent of nowhere differentiable character, it turns out that each function in $\mathcal {P}$ is nowhere differentiable. The class $\mathcal {P}$ naturally appears from both a geometrical viewpoint and an analytic viewpoint. In fact, we prove that a function belongs to $\mathcal {P}$ if and only if some geometrical inequality holds for a family of parabolas with vertexes on this function. As its application, we study the behavior of the Hamilton–Jacobi flow starting from a function in $\mathcal {P}$. A connection between $\mathcal {P}$ and some functional series is also investigated. In terms of second-order central differences, we give a necessary and sufficient condition so that a function given by the series belongs to $\mathcal {P}$. This enables us to construct a large number of examples of functions in $\mathcal {P}$ through an explicit formula.