Constructions of the (global) fractal interpolation functions on standard function spaces got a lot of attention in the last centuries. Motivated by the newly introduced local fractal functions corresponding to a local iterated functions system which is the generalization of the traditional iterated functions system we construct the local non-affine α - fractal functions in this article. A few examples of the graphs of these functions are provided. A fractal operator which takes the classical function to its local fractal counterpart is defined and some of its properties are also studied.
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