Quantum α-Fractal Approximation

Fractal approximation is a well studied concept, but the convergence of all the existing fractal approximants towards the original function follows usually if the magnitude of the corresponding scaling factors approaches zero. In this article, for a given function (Formula presented.) by exploiting fractal approximation theory and considering the classical q-Bernstein polynomials as base functions, we construct a sequence (Formula presented.) of (Formula presented.) -fractal functions that converges uniformly to f for any choice of the scaling facuntions/ scaling factors. The convergence of the sequence (Formula presented.) of (Formula presented.) -fractal functions towards f follows from the convergence of the sequence of q-Bernstein polynomials of f towards f. If we consider a sequence (Formula presented.) of positive functions on a compact real interval that converges uniformly to a function f, we develop a double sequence (Formula presented.) of (Formula presented.) -fractal functions that converges uniformly to f.