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

• Published in: Acta mathematica Hungarica Vol. 160; no. 2; pp. 343 - 359
• Main Authors: Fujita, Y., Hamamuki, N., Siconolfi, A., Yamaguchi, N.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-04-2020; Springer Nature B.V
• Subjects: Concavity; Continuity (mathematics); Mathematics; Mathematics and Statistics; Parabolas; Periodic functions; the Takagi function; primary 26A27; geometric inequality; secondary 39B22; nowhere differentiable function; inf-convolution
• ISSN: 0236-5294; 1588-2632
• DOI: 10.1007/s10474-019-01007-3

We introduce and investigate a class
P
of continuous and periodic functions on
R
. The class
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
P
is nowhere differentiable. The class
P
naturally appears from both a geometrical viewpoint and an analytic viewpoint. In fact, we prove that a function belongs to
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
P
. A connection between
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
P
. This enables us to construct a large number of examples of functions in
P
through an explicit formula.