Narrow Results

In this paper, we study solutions of a variation of a classical integral equation (based on the Picard operator) in which Lebesgue measure is replaced by a self-similar measure $\mu$. Our main interest is in the fractal nature of the solutions and we use Iterated Function System (IFS) tools to investigate the behavior and self-similarity of these solutions. Both the integral and differential forms of the equation are discussed since each brings useful insights. Several convergence results are provided along with illustrative examples that show the applications of the theory when the underlying fractal object is the celebrated Cantor set. Additionally we show that the solution to our integral equation inherits self-similarity from the defining measure $\mu$.

Subdivision schemes generate self-similar curves and surfaces for which it has a familiar connection between fractal curves and surfaces generated by iterated function systems (IFS). Overveld [Comput.-Aided Des. 22(9) (1990) 591–597] proved that the subdivision matrices can be perturbated in such a way that it is possible to get fractal-like curves that are perturbated Bézier cubic curves. In this work, we extend the Overveld scheme to $n$th degree curves, and deduce the condition for curvature continuity and convex hull property. We find the conditions for positive preserving fractal-like Bézier curves in the proposed subdivision matrices. The resulting 2D/3D curves from these binary subdivision matrices resemble with fractal images. Finally, the dependence of the shape of these fractal-like curves on the elements of subdivision matrices is demonstrated with suitably chosen examples.

In this paper, we mainly investigate continuous functions with unbounded variation on closed intervals. Given the increasing number of proposals and definitions of different kinds of functions with fractal structure in the scope of fractal functions, it is important to introduce a systematic classification and discuss elementary definitions of fractal functions. Different examples of fractal functions with certain fractal dimensions have been given. Then, we have introduced different kind of fractal function such as singular fractal function, irregular fractal function, regular fractal function and complete regular fractal function. We also introduce fractal functions of integer dimension with one and two, respectively. Other continuous and discontinuous functions with fractal structure have been given. Applications of fractal functions in interpolation, approximation and relationship between fractional calculus have been investigated. Finally, we studied fractal dimension spaces with different fractal dimensions.

This paper investigates fractal dimension of product of continuous functions with Box dimension on $[0,1]$. For two continuous functions with different Box dimensions, the Box dimension of their product has been proved to be the larger one. Furthermore, the Box dimension of product of two continuous functions with the same Box dimension may not exist. Definitions of regular fractal and local fractal functions have been given. Product of a regular fractal function and a local fractal function with the same Box dimension must still be the original Box dimension.

This expository paper exhibits and discusses the communalities and differences between wavelets and fractal functions, both examples of multiscale processes.