Approximation properties of bivariate α-fractal functions and dimension results

ABSTRACT

In this paper, we study a different class of bivariate α-fractal functions. First, we introduce the bivariate Bernstein α-fractal functions that are more suitable to approximate both smooth and non-smooth surfaces and investigate their convergence properties. Then, we compute the box-counting dimension of the graph of the bivariate α-fractal functions for equally spaced data set. In regard to the connection of functional analysis and fractal function, we cogitate the bivariate fractal operator in spaces of functions such as k-times continuously differentiable functions space ${\mathcal{C}}^k(I\times J)$ and the Lebesgue space $L^p(I\times J)$. Also, we study some approximation properties using bivariate Bernstein α-fractal trigonometric functions.

Keywords:

• Fractal function
• box dimension
• convergence
• function space
• trigonometric approximation

AMS CLASSIFICATIONS:

• 28A80
• 46B25
• 42A15
• 42A20
• 65T40

Acknowledgments

We thank the anonymous reviewers for the valuable and constructive suggestions that helped to improve the manuscript.

Disclosure statement

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