Some Functional Equations Revisited: The Multifractal Properties

Journal Article

2003

Published in: http://www.tandfonline.com/doi/abs/10.1080/1065246031000074335

We review the solutions of some functional equations for which the differentiability was the aim of study in the beginning of the last century. By decomposing these functions on the Schauder basis (which are only Lipschitz of order 1), we determine the exact Hölder regularity (even when it exceeds 1) and thus prove that this regularity changes widely from point to point. We also determine the Sobolev (or Besov) spaces to which these functions belong. We finally study the validity of the multifractal formalism which relates some (Sobolev) functional norms of a function to its 'Hölder spectrum' (which is the dimension of the set of points where the function has a given Hölder regularity).

• Volume Number: Vol. 14, No. 4, August 2003
• Issue Number: Vol. 14, No. 4, August 2003
• Magazine \ Newspaper: Integral Transforms and Special Functions
• Pages: 333–348