A class of nowhere differentiable functions satisfying some concavity-type estimate
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 th...
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| Published in: | Acta mathematica Hungarica Vol. 160; no. 2; pp. 343 - 359 |
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| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Cham
Springer International Publishing
01-04-2020
Springer Nature B.V |
| Subjects: | |
| Online Access: | Get full text |
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| Summary: | 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. |
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| ISSN: | 0236-5294 1588-2632 |
| DOI: | 10.1007/s10474-019-01007-3 |