Constrained Fractal Trigonometric Approximation: A Sequential Approach

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In this paper, for a given \(f\in \mathcal {C}(I),\) we construct a sequence of Bernstein \(\alpha \)-fractal functions with variable scaling factors. Bernstein \(\alpha \)-fractal functions proposed in this paper are fixed points of the Read-Bajraktarevic operators defined on a suitable function space. The convergence of the Bernstein \(\alpha \)-fractal function with variable scaling factors towards the original function f follows from the convergence of Bernstein polynomials of f toward f. Owing this reason, Bernstein \(\alpha \)-fractal function with variable scaling factors proposed in this paper converge to the original function for any choice of the variable scalings whereas the existing fractal approximants converge to the original function only if the magnitude of the scaling factors goes to zero. Using the Bernstein \(\alpha \)-fractal functions, we study the Bernstein \(\alpha \)-fractal trigonometric polynomials with variable scaling factors and their constrained approximation properties. For a given sequence \(\{f_n(x)\}^{\infty }_{n=1}\) of continuous periodic functions defined on \([-l,l]\) that converges uniformly to a function f, we investigate the existence of a double sequence of Bernstein \(\alpha \)-fractal trigonometric polynomials that converges uniformly to f.