Fractal curves and wavelets

V. Yu. Protasov

Izvestiya. Mathematics , Tome 70 (2006) no. 5, pp. 975-1013

Résumé

We introduce the notion of a summable fractal curve generated by a finite family of affine operators. This generalizes well-known notions of affine fractals and continuous fractal curves to the case of non-contractive operators. We establish a criterion for the existence of a fractal curve for a given family of operators, obtain criteria for that curve to belong to various function spaces and derive formulae for the exponents of regularity in those spaces as well as asymptotically sharp estimates for the moduli of continuity. These results are applied to the study of well-known curves (Koch, de Rham, and so on), refinable functions and wavelets. We also study the local behaviour of continuous fractal curves. We obtain a formula for the exponent of local regularity of continuous fractal curves at a given point and characterize the sets of points with a fixed local regularity. It is shown that the values of the local regularity of any fractal curve fill out some closed interval. Nevertheless, the regularity is the same at almost all points (in the Lebesgue measure) and can be computed from the Lyapunov exponent of certain linear operators. We apply this technique to refinement equations and compactly supported wavelets. As an example, we compute the moduli of continuity and exponents of local regularity and $L_p$-regularity for several Daubechies wavelets.