Construction of Fractal Bases for Spaces of Functions.

TL;DR: The construction of fractal versions of classical functions as polynomials, trigonometric maps, etc by means of a particular Iterated Function System of the plane is tackled because of the closeness between the classical function and its fractal analogue.

Abstract: The construction of fractal versions of classical functions as polynomials, trigonometric maps, etc. by means of a particular Iterated Function System of the plane is tackled. The closeness between the classical function and its fractal analogue provides good properties of approximation and interpolation to the latter. This type of methodology opens the use of non-smooth and fractal functions in approximation. The procedure involves the definition of an operator mapping standard functions into their dual fractals. The transformation is linear and bounded and some bounds of its norm are established. Through this operator we define families of fractal functions that generalize the classical Schauder systems of Banach spaces and the orthonormal bases of Hilbert spaces. With an appropriate election of the coefficients of Iterated Function System we define sets of fractal maps that span the most important spaces of functions as \({\mathcal {C}}[a,b]\) or \({\mathcal {L}}^p[a,b]\).