Quantum α-fractal approximation

Abstract

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 $f\in \mathcal{C}(I),$ by exploiting fractal approximation theory and considering the classical q-Bernstein polynomials as base functions, we construct a sequence $\{f_n^{(q,\alpha )}(x){\}}_{n=1}^{\mathrm{\infty }}$ of $(q,\alpha )$-fractal functions that converges uniformly to f even if the norm/magnitude of the scaling functions/scaling factors does not tend to zero. The convergence of the sequence $\{f_n^{(q,\alpha )}(x){\}}_{n=1}^{\mathrm{\infty }}$ of $(q,\alpha )$-fractal functions towards f follows from the convergence of the sequence of q-Bernstein polynomials of f towards f. If we consider a sequence $\{f_m(x){\}}_{m=1}^{\mathrm{\infty }}$ of positive functions on a compact real interval that converges uniformly to a function f, we develop a double sequence $\{\{f_{m,n}^{(q,\alpha )}(x){\}}_{n=1}^{\mathrm{\infty }}{\}}_{m=1}^{\mathrm{\infty }}$ of $(q,\alpha )$-fractal functions that converges uniformly to f.

Keywords:

• q-Bernstein polynomials
• fractal functions
• constrained approximation
• fractals
• interpolation

2010 Mathematics Subject Classifications:

• 28A80
• 26C15
• 26A48
• 26A51
• 65D05

Acknowledgements

The first author acknowledges the financial support received from Council of Scientific & Industrial Research (CSIR), India (Project No. 25(0290)/18/EMR-II).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author acknowledges the financial support received from Council of Scientific & Industrial Research (CSIR), India (Project No. 25(0290)/18/EMR-II).