Fractal Dimensions and Divergence of Fourier Series

Abstract

We show that for any 1 ≤ s ≤ 2, there is a periodic continuous function f whose Fourier series is divergent at some point, and whose graph satisfies

{\text{dim}}_H(\text{graph}f)={\text{dim}}_B(\text{graph}f)=s.

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text{dim}_{H}(\text{graph} \ f)=\text{dim}_{B}(\text{graph} \ f)=s.$$\end{document}

Keywords

Fractal dimensions / Fourier series / 28A80

References

[1] Barański K. On the dimension of graphs of Weierstrass-type functions with rapidly growing frequencies. Nonlinearity, 2012, 25(1): 193-209

[2] Bary N. A Treatise on Trigonometric Series, Vols. I, II, 1964, New York, The Macmillan Company

[3] Duoandikoetxea J. Fourier Analysis, 2001, Providence, RI, American Mathematical Society29

[4] Falconer K. Fractal Geometry—Mathematical Foundations and Applications, 2003Second EditionHoboken, NJ, John Wiley & Sons

[5] Grafakos L. Classical Fourier Analysis, 2008Second EditionNew York, Springer 249

[6] Humke PD, Petruska G. The packing dimension of a typical continuous function is 2. Real Anal. Exchange, 1988, 14(2): 345-358 89

[7] Hyde J, Laschos V, Olsen L, Petrykiewicz I, Shaw A. On the box dimensions of graphs of typical continuous functions. J. Math. Anal. Appl., 2012, 391(2): 567-581

[8] Mattila P. Fourier Analysis and Hausdorff Dimension, 2015, Cambridge, Cambridge University Press 150

[9] Stein E, Shakarchi R. Fourier Analysis—An Introduction, 2003, Princeton, NJ, Princeton University Press1

RIGHTS & PERMISSIONS

Peking University