Local fractal functions and function spaces

We introduce local iterated function systems and present some of their basic properties. A new class of local attractors of local iterated function systems, namely local fractal functions, is constructed. We derive formulas so that these local fractal functions become elements of various function spaces, such as the Lebesgue spaces $L^p$, the smoothness spaces $C^n$, the homogeneous H\"older spaces $\dot{C}^s$, and the Sobolev spaces $W^{m,p}$.