The Takagi function: a survey

DOI10.14321/realanalexch.37.1.0001zbMath1248.26007arXiv1110.1691OpenAlexW2109509083MaRDI QIDQ418203

Kiko Kawamura, Pieter C. Allaart

Publication date: 28 May 2012

Published in: Real Analysis Exchange (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1110.1691

zbMATH Keywords

functional equationmodulus of continuityHausdorff dimensionlevel setTakagi functionvan der Waerden functioncontinuous nowhere differentiable functioninfinite derivativesLebesgue's singular function

Mathematics Subject Classification ID

Lipschitz (Hölder) classes (26A16) Fractals (28A80) Hausdorff and packing measures (28A78) Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives (26A27)

Related Items (54)

Step roots of Littlewood polynomials and the extrema of functions in the Takagi class ⋮ Limiting curves for the Pascal adic transformation ⋮ Unnamed Item ⋮ Non-differentiability and Hölder properties of self-affine functions ⋮ Second order differentiability and related topics in the Takagi class ⋮ Edge-Isoperimetric Problem for Cayley Graphs and Generalized Takagi Functions ⋮ Superdifferential analysis of the Takagi-van der Waerden functions ⋮ Weak tangent and level sets of Takagi functions ⋮ No-where averagely differentiable functions: Baire category and the Takagi function ⋮ A class of nowhere differentiable functions satisfying some concavity-type estimate ⋮ On the minimum value of the Colless index and the bifurcating trees that achieve it ⋮ Asymptotic expansions for linear homogeneous divide-and-conquer recurrences: algebraic and analytic approaches collated ⋮ Approximate differentiability in the generalized Takagi-van der Waerden class ⋮ Geometrical properties of the mean-median map ⋮ On global extrema of power Takagi functions ⋮ Identities and periodic oscillations of divide-and-conquer recurrences splitting at half ⋮ Level sets of the Takagi function: local level sets ⋮ Global extrema of the Delange function, bounds for digital sums and concave functions ⋮ Incidence problems in harmonic analysis, geometric measure theory, and ergodic theory. Abstracts from the workshop held June 4--9, 2023 ⋮ A new proof of the approximate convexity of the Takagi function ⋮ Limiting curves for the dyadic odometer ⋮ Dimension of the repeller for a piecewise expanding affine map ⋮ Hamilton-Jacobi flows with nowhere differentiable initial data ⋮ Properties of functions with monotone graphs ⋮ On the $p$th variation of a class of fractal functions ⋮ Differentiability of continuous functions in terms of Haar-smallness ⋮ Fractional integrals, derivatives and integral equations with weighted Takagi–Landsberg functions ⋮ All the generalized characteristics for the solution to a Hamilton-Jacobi equation with the initial data of the Takagi function ⋮ Unnamed Item ⋮ An Amusing Sequence of Functions ⋮ Application of extreme sub- and epiarguments, convex and concave envelopes to search for global extrema ⋮ Number theoretic applications of a class of Cantor series fractal functions. I ⋮ Level sets of signed Takagi functions ⋮ On (signed) Takagi-Landsberg functions: \(p\)th variation, maximum, and modulus of continuity ⋮ On a class of generalized Takagi functions with linear pathwise quadratic variation ⋮ Differentiability of the functions of the generalized Takagi class ⋮ On the level sets of the Takagi-van der Waerden functions ⋮ Pointwise Hölder exponents of the complex analogues of the Takagi function in random complex dynamics ⋮ Non approximate derivability of the Takagi function ⋮ Unnamed Item ⋮ Hölder exponents of self-similar functions ⋮ The Kruskal-Katona function, Conway sequence, Takagi curve, and Pascal adic ⋮ On the rate of convergence for Takagi class functions ⋮ Random deviations of ergodic sums for the Pascal adic transformation in the case of the Lebesgue measure ⋮ Dimension theory of some non-Markovian repellers. II: Dynamically defined function graphs ⋮ Functions with antiderivative. I: Characterizations and properties ⋮ Analysis of a fractal boundary: the graph of the Knopp function ⋮ Infinite derivatives of the Takagi-Van der Waerden functions ⋮ Fractal random series generated by Poisson-Voronoi tessellations ⋮ Hausdorff dimension of local level sets of Takagi's function ⋮ A limit theorem for Bernoulli convolutions and the \(\Phi \)-variation of functions in the Takagi class ⋮ Sierpinski-Takagi combination for a uniform and optimal point-surface load transmission ⋮ A self-affine property of evolutional type appearing in a Hamilton-Jacobi flow starting from the Takagi function ⋮ Nonlinear approximation and (deep) ReLU networks

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• On the Karaś type theorems for the multidegrees of polynomial automorphisms
• An inequality for sums of binary digits, with application to Takagi functions
• The Hausdorff dimension of the level sets of Takagi's function
• On generalized Takagi functions
• On the set of points where Lebesgue's singular function has the derivative zero
• Sur un exemple de fonction continue sans dérivée
• Sur l'exemple, donné par M. de Rham, d'une fonction continue sans dérivée
• On the Hölder continuity of certain functions
• A generalization of Hata-Yamaguti's results on the Takagi function
• On probabilistic properties of Takagi's function
• Level sets of the Takagi function: Hausdorff dimension
• Approximate convexity of Takagi type functions
• The improper infinite derivatives of Takagi's nowhere-differentiable function
• On a flexible class of continuous functions with uniform local structure
• Takagi functions and approximate midconvexity
• Distribution of the maxima of random Takagi functions
• Irregular \(1\)-sets on the graphs of continuous functions
• Power sums of digital sums
• On modified Takagi functions of two variables
• Sur la fonction sommatoire de la fonction 'somme des chiffres'
• Minimum shadows in uniform hypergraphs and a generalization of the Takagi function
• On the classification of self-similar sets determined by two contractions on the plane.
• On the random series \(\sum\pm\lambda^ n\) (an Erdös problem)
• An explicit formula of the exponential sums of digital sums
• The Mcfunction. [The Takagi function]
• Slow points for functions in the Zygmund class \(\Lambda^*_d\)
• Smooth functions
• On Maxima of Takagi-van der Waerden Functions
• Van Der Waerden's Continuous Nowhere Differentiable Function
• On Van Der Waerden's Nowhere Differentiable Function
• On the Hausdorff Dimension of Some Graphs
• The Takagi function and its generalization
• On the Continuity of Van Der Waerden's Function in the Holder Sense
• Power and Exponential Sums of Digital Sums Related to Binomial Coefficient Parity
• A Survey of Combinatorial Gray Codes
• Smooth Interpolation, Holder Continuity, and the Takagi-van der Waerden Function
• Micro tangent sets of continuous functions
• Probabilistic Behaviour of Functions in the Zygmund Spaces ∧ ^* and λ ^*
• Sur la théorie de la mesure dans les espaces combinatoires et son application au calcul des probabilités. I. Variables indépendantes
• Digital Sum Problems for the Gray Code Representation of Natural Numbers
• ON APPROXIMATELY MIDCONVEX FUNCTIONS
• The Takagi Function and Its Properties
• An inequality for the Takagi function
• On Zeros of a Continuous Nowhere Differentiable Function
• An Explicit Expression for Binary Digital Sums
• On A Function of Van Der Waerden
• Sets of Fractional Dimensions (V): on Dimensional Numbers of Some Continuous Curves
• On Some Singular Monotonic Functions Which Are Strictly Increasing
• Extreme values of some continuous nowhere differentiable functions

This page was built for publication: The Takagi function: a survey