The Takagi function: a survey
By P. Allaart, K. Kawamura. 2011
130 citations recorded by the source corpus.
More than a century has passed since Takagi [75] published his simple example of a continuous but nowhere differentiable function, yet Takagi’s function – as it is now commonly referred to despite repeated rediscovery by mathematicians in the West – continues to inspire, fascinate and puzzle researchers as never before. For this reason, and also because we have noticed that many aspects of the Takagi function continue to be rediscovered with alarming frequency, we feel the time has come for a comprehensive review of the literature. Our goal is not only to give an overview of the history and known characteristics of the function, but also to discuss some of the fascinating applications it has found – some quite recently! – in such diverse areas of mathematics as number theory, combinatorics, and analysis. We also include a section on generalizations and variations of the Takagi function. In view of the overwhelming amount of literature, however, we have chosen to limit ourselves to functions based on the “tent map”. In particular, this paper shall not make more than a passing mention of the Weierstrass function and is not intended as a general overview of continuous nowhere-differentiable functions. We thank Prof. Paul Humke for encouraging us to write this survey, and for issuing periodic cheerful reminders.
Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/438310e4c228e1d43cb0f7f464dec0a5738a4948
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Citations in the corpus
- The Takagi function and its properties (Functions in Number Theory and Their Probabilistic Aspects) (2011, result): “Allaaert and Kawamura [4] further study the extreme values of functions related to the Takagi function.”
- Assouad Dimension and Fractal Geometry (2020, background): “The resulting functions are known as Takagi functions and are easier to handle than the Weierstrass function, see the survey [3], and the Hausdorff, box and packing dimensions are also known to be given by the formula (17.”