Differentiability and Dimension of Some Fractal Fourier Series

  title={Differentiability and Dimension of Some Fractal Fourier Series},
  author={Fernando Chamizo and Antonio Manj{\'o}n-Cabeza C{\'o}rdoba},
  journal={Advances in Mathematics},
Taking into account that the frequencies of Fourier expansions are integers it is not surprising that some arithmetical results play an important role in many theorems and examples in harmonic analysis. This relation is even clearer from the historical point of view and Hardy and Littlewood can be considered at the same time as founders of a substantial part of harmonic analysis and analytic number theory. For instance, in [Ha-Li] they considered 

Figures from this paper

Automorphic forms and differentiability properties

We consider Fourier series given by a type of fractional integral of automorphic forms, and we study their local and global properties, especially differentiability and fractal dimension of the graph

Multifractal behavior of polynomial Fourier series

Some Fourier series with gaps

We examine diverse local and global aspects of the family of Fourier series ∑n−αe(nkx). In particular, combining number theoretical and harmonic analytic arguments, we study differentiability, Hölder

Fractal Curves from Prime Trigonometric Series

We study the convergence of the parameter family of series: V α , β ( t ) = ∑ p p − α exp ( 2 π i p β t ) , α , β ∈ R > 0 , t ∈ [ 0 , 1 ) defined over prime numbers p and, subsequently, their

Asymptotic behaviour and Hausdorff dimension of Riemann's non-differentiable function

Riemann's non-differentiable function, whose analytic regularity has been widely studied, can also be analysed from a geometric perspective. Indeed, it can be generalised to the complex plane to

H\"older continuity and dimensions of fractal Fourier series

. Motivated by applications in number theory, analysis, and fractal geometry, we consider regularity properties and dimensions of graphs associated with Fourier series of the form F ( t ) = (cid:80)

Fractal solutions of dispersive partial differential equations on the torus

We use exponential sums to study the fractal dimension of the graphs of solutions to linear dispersive PDE. Our techniques apply to Schrödinger, Airy, Boussinesq, the fractional Schrödinger, and the

Some geometric properties of Riemann's non-differentiable function

On the Hausdorff dimension of Riemann’s non-differentiable function

Recent findings show that the classical Riemann's non-differentiable function has a physical and geometric nature as the irregular trajectory of a polygonal vortex filament driven by the binormal

An Analytical Study of Flatness and Intermittency through Riemann's Nondifferentiable Functions

. In the study of turbulence, intermittency is a measure of how much Kolmogorov’s theory of 1941 deviates from experimental measurements. It is quantified with the flatness of the velocity of the fluid,



Pointwise analysis of Riemann's “nondifferentiable” function

SummaryWe will show how to analyse the local regularity of functions with the help of the wavelet transform. These results will be applied to the function of Riemann, where we show the existence of a

Selfsimilarity of "Riemann's nondifferentiable function"

This is an expository article about the series f(x) = 1 X n=1 1 n 2 sin(n 2 x); which according to Weierstrass was presented by Riemann as an example of a continuous function without a derivative. An


  • J. Gerver
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1969
It is shown that a continuous function which Riemann is said to have believed to be nowhere differentiable is in fact differentiable at certain points.

Multiplicative Number Theory

From the contents: Primes in Arithmetic Progression.- Gauss' Sum.- Cyclotomy.- Primes in Arithmetic Progression: The General Modulus.- Primitive Characters.- Dirichlet's Class Number Formula.- The

An Introduction to the Theory of Numbers

This is the fifth edition of a work (first published in 1938) which has become the standard introduction to the subject. The book has grown out of lectures delivered by the authors at Oxford,

Graduate Texts in Mathematics

Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully

Le grand crible dans la théorie analytique des nombres

© Société mathématique de France, 1974, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les

Über Continuirliche Functionen Eines Reellen Arguments, die für Keinen Werth des Letzteren Einen Bestimmten Differentialquotienten Besitzen

Bis auf die neueste Zeit hat man allgemein angenommen, dass eine eindeutige und continuirliche Function einer reellen Veranderlichen auch stets eine erste Ableitung habe, deren Werth nur an einzelnen