Haar wavelets of higher order on fractals and regularity of functions

Wavelets of Haar type of higher order m on self-similar fractals were introduced by the author in J. Fourier Anal. Appl. 4 (1998) 329–340. These are piecewise polynomials of degree m instead of piecewise constants. It was shown that for certain totally disconnected fractals, spaces of functions defined on the fractal may be characterized by means of the magnitude of the wavelet coefficients of the functions. In this paper, the study of these wavelets is continued. It is shown that also in the case when the fractals are not totally disconnected, the wavelets can be used to study regularity properties of functions. In particular, the self-similar sets considered can be, e.g., an interval in R or a cube in Rn. It turns out that it is natural to use Haar wavelets of higher order also in these classical cases, and many of the results in the paper are new also for these sets.  2003 Elsevier Inc. All rights reserved.