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We introduce local iterated function systems (IFSs) and present some of their basic properties. A new class of local attractors of local IFSs, 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 Holder spaces \(\dot{C}^{s}\), and the Sobolev spaces W m, p .

Local Fractal Functions and Function Spaces