Generalization of the Piecewise Polynomial Interpolation by Fractal Functions

The fractal interpolation functions defined by iterated function systems provide new methods of approximation and quantification of experimental data. The polynomial fractal functions can be considered as generalization of the piecewise polynomial interpolants. Assuming some hypotheses on the original function, a bound of the representation of the error for this kind of approximants is obtained here. The results proved guarantee the convergence of the interpolant to any smooth function when the diameter of the partition approaches zero. The property of good fit of the derivatives is also verified if the iterated function system is adequately chosen.