Acta Applicandae Mathematicae

, Volume 159, Issue 1, pp 11–27 | Cite as

Bernstein Fractal Trigonometric Approximation

  • N. VijenderEmail author


Fractal interpolation and approximation received a lot of attention in the last thirty years. The main aim of the current article is to study a fractal trigonometric approximants which converge to the given continuous function even if the magnitude of the scaling factors does not approach zero. In this paper, we first introduce a new class of fractal approximants, namely, Bernstein \(\alpha \)-fractal functions using the theory of fractal approximation and Bernstein polynomial. Using the proposed class of fractal approximants and imposing no condition on corresponding scaling factors, we establish that the set of Bernstein \(\alpha \)-fractal trigonometric functions is fundamental in the space of continuous periodic functions. Fractal version of Gauss formula of trigonometric interpolation is obtained by means of Bernstein trigonometric fractal polynomials. We study the Bernstein fractal Fourier series of a continuous periodic function \(f\) defined on \([-l,l]\). The Bernstein fractal Fourier series converges to \(f\) even if the magnitude of the scaling factors does not approach zero. Existence of the \(\mathcal{C}^{r}\)-Bernstein fractal functions is investigated, and Bernstein cubic spline fractal interpolation functions are proposed based on the theory of \(\mathcal{C}^{r}\)-Bernstein fractal functions.


\(\alpha \)-Fractal functions Bernstein polynomials Fractal approximation Convergence Fractal trigonometric approximation Fractal Fourier series 

Mathematics Subject Classification

30E10 28A80 41A30 41A17 65D017 



The author would like to thank the anonymous reviewers for helpful and constructive comments that greatly contributed to improve the quality and presentation of the manuscript.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Mathematics Division, School of Advanced SciencesVIT University ChennaiChennaiIndia

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